BBP-Type Formula

Author

Eric W. Weisstein
September 29, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/BBP-TypeFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BBP-TypeFormula.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Utilities`Simplify`

Usage

FindBasis[Pi,16,8]

FindBasis[Pi,16,8,BasisPower->1]

BasisSum[#,k]&/@%//ColumnForm//TraditionalForm

FindBasis[Pi^2 ,64,6,BasisPower->2]

BasisSum[#,k]&/@%//ColumnForm//TraditionalForm

BasisSum[#,k]&/@FindBasis[Catalan ,-1,2,BasisPower->2]//ColumnForm//TraditionalForm

BasisSum[#,k]&/@FindBasis[Catalan ,-1,10,BasisPower->2]//ColumnForm//TraditionalForm

Discoveries

0

0

2

vecs2={{16,8,{1.1200709999999958` Second,{{0,{8,-8,-4,-8,-2,-2,1,0,0},16,8,1}}}},
{16,16,{3.844240999999954` Second,{{0,{0,8,0,-8,0,-4,0,-8,0,-2,0,-2,0,1,0,0,0},16,16,1}}}},
{16,24,{9.756609999999966` Second,{{0,{0,0,8,0,0,-8,0,0,-4,0,0,-8,0,0,-2,0,0,-2,0,0,1,0,0,0,0},16,24,1},{0,{8,0,0,-32,0,0,4,0,0,4,0,0,-2,0,0,-12,0,0,-1,0,0,1,0,0,0},16,24,1},{0,{0,16,0,0,-8,0,0,-48,0,0,-4,0,0,4,0,0,2,0,0,-8,0,0,1,0,0},16,24,1}}}},
{64,6,{0.7560470000000805` Second,{{0,{16,-24,-8,-6,1,0,0},64,6,1}}}},
{64,12,{2.4241519999999355` Second,{{0,{0,16,0,-24,0,-8,0,-6,0,1,0,0,0},64,12,1}}}},
{64,18,{5.256329000000051` Second,{{0,{0,0,16,0,0,-24,0,0,-8,0,0,-6,0,0,1,0,0,0,0},64,18,1}}}},
{64,24,{10.180634999999825` Second,{{0,{0,0,0,16,0,0,0,-24,0,0,0,-8,0,0,0,-6,0,0,0,1,0,0,0,0,0},64,24,1}}}},
{256,16,{3.8242389999999205` Second,{{0,{128,-128,-64,-128,-32,-32,16,0,8,-8,-4,-8,-2,-2,1,0,0},256,16,1}}}},
{4096,12,{2.1881369999999993` Second,{{0,{1024,-1536,-512,-384,64,0,16,-24,-8,-6,1,0,0},4096,12,1}}}},
{4096,24,{8.29651800000002` Second,{{0,{0,0,0,0,-256,256,128,0,128,-128,-64,-64,0,-16,0,24,4,-4,-2,-2,-2,-3,1,0,0},4096,24,1},{0,{256,-768,256,0,64,64,64,96,0,16,0,-16,-4,-16,-4,-6,-1,-1,-1,1,0,0,0,0,0},4096,24,1},{0,{256,256,-256,-512,-192,-448,-64,-288,-64,-48,0,-16,-4,16,4,-6,3,-1,1,-3,1,0,0,0,0},4096,24,1},{0,{-256,256,768,-256,64,-192,0,-288,-64,16,0,-16,4,0,-12,-18,-1,-5,0,-1,1,0,0,0,0},4096,24,1},{0,{256,-256,-256,-768,192,192,128,288,128,112,0,16,-4,-16,4,-6,-3,-3,-2,-1,-2,1,0,0,0},4096,24,1}}}},
{262144,18,{5.072317000000055` Second,{{0,{65536,-98304,-32768,-24576,4096,0,1024,-1536,-512,-384,64,0,16,-24,-8,-6,1,0,0},262144,18,1}}}},
{531441,24,{16.739999999999945` Second,{{0,{0,-59049,118098,0,-19683,-26244,0,-6561,-4374,-2916,729,0,0,-81,162,0,-27,-36,0,-9,-6,-4,1,0,0},531441,24,1},{0,{-177147,236196,118098,59049,0,26244,6561,0,-4374,729,0,0,-243,324,162,81,0,36,9,0,-6,1,0,0,0},531441,24,1}}}}};

Length[#[[3,2]]]&/@vecs2

sums2=Map[BasisSum[#,k]&,#[[-1,-1]]&/@vecs2,{2}]

N[ReleaseHold[sums2],100]

3

vecs3=Flatten[#[[-1,-1]]&/@{{729,12,{4.070313999999996` Second,{{0,{0,-81,162,0,-27,-36,0,-9,-6,-4,1,0,0},729,12,1},{0,{-243,324,162,81,0,36,9,0,-6,1,0,0,0},729,12,1}}}},
{729,24,{23.24902300000008` Second,{{0,{0,0,0,-81,0,162,0,0,0,-27,0,-36,0,0,0,-9,0,-6,0,-4,0,1,0,0,0},729,24,1},{0,{0,-243,0,324,0,162,0,81,0,0,0,36,0,9,0,0,0,-6,0,1,0,0,0,0,0},729,24,1}}}}},1];

sums3=BasisSum[#,k]&/@vecs3

ReleaseHold[sums3]

Log

ln 2==(2/3) (∑_(k=0)^∞)(1/(9^k))(( 1)/(2 k+1))

(2/3)Sum[(1/((9^k)(2k+1))),{k,0,∞}]

FullSimplify[%]

FindBasis[Log[2],9,2]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

Log[2]

(((-2))^k)

(((-1))^k)

Do[Print[{-1,k,FindBasis[Log[2],-1,k]//Timing}],{k,48}]

(2^k)

(3^k)

((Log[2])^2)

(ln^2)2==(1/32) (∑_(k=0)^∞)(1/(64^k)) (-(160/(((6 k+2))^2))-(56/(((6 k+3))^2))-(40/(((6 k+4))^2))+(4/(((6 k+5))^2))-(1/(((6 k+6))^2))+(64/(((6 k+1))^2)))

Sum[(1/32)(1/(64^k))((64/(((6k+1))^2))-(160/(((6k+2))^2))-(56/(((6k+3))^2))-(40/(((6k+4))^2))+(4/(((6k+5))^2))-(1/((6k+6)^2))),{k,0,∞}]

fb=FindBasis[((Log[2])^2),64,6,2]

b=BasisSum[#,k]&/@fb

FullSimplify[ReleaseHold[b]]

N[%]

Log[3]

ln 3==(∑_(k=0)^∞)(1/(4^k))(1/( 2 k+1))

Sum[(1/((4^k)(2k+1))),{k,0,∞}]

FullSimplify[%]

FindBasis[Log[3],4,2]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

ln 3==(1/729) (∑_(k=0)^∞)(1/(729^k)) ((81/(6 k+2))+(81/(6 k+3))+(9/(6 k+4))+(9/(6 k+5))+(1/(6 k+6))+(729/(6 k+1)))

(1/729)Sum[(1/(729^k))Plus@@MapIndexed[((#1)/(6k+#2[[1]]))&,{729,81,81,9,9,1}],{k,0,∞}]

FullSimplify[%]

FindBasis[Log[3],729,6]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

ln((9/10))

FullSimplify[-Sum[(1/(k (10^k))),{k,1,∞}]]

FullSimplify[Subtract@@{-Log[(10/9)],Log[9/10]}]

fb=FindBasis[Log[9/10],10,1]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

ln((1111111111/387420489))

(1/(10^8))Sum[(1/(10^(10k)))Plus@@MapIndexed[((#1)/(10k+#2[[1]]))&,10^Range[8,0,-1]],{k,0,∞}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

fb=FindBasis[Log[(1111111111/387420489)],(10^10),10]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

(25/2)ln((781/256)(((((57-5Sqrt[5])/(57+5Sqrt[5]))))^(Sqrt[5])))

Sum[(1/(5^(5k)))((5/(5k+2))+(1/(5k+3))),{k,0,∞}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

FullSimplify[Subtract@@{(25/2) Log[(781/256) (((((57-5 (Sqrt[5]))/(57+5 (Sqrt[5])))))^(Sqrt[5]))],(25/2) (Log[(781/256)]+(Sqrt[5]) Log[-1+(114/(57+5 (Sqrt[5])))])}]

fb=FindBasis[Log[(781/256) (((((57-5 (Sqrt[5]))/(57+5 (Sqrt[5])))))^(Sqrt[5]))],(5^5),5]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

π

(((-1))^k)

l={{{π,{-4,0,1},-1,2,1}},
{{π,{-3,0,0,0,-3,0,1},-1,6,1}},
{{π,{-4,0,4,0,-4,0,4,0,-4,0,1},-1,10,1}},{{π,{-3,0,3,0,-3,0,-4,0,-3,0,3,0,-3,0,1},-1,14,1}},{{π,{-2,0,-3,0,-2,0,2,0,0,0,2,0,-2,0,-3,0,-2,0,1},-1,18,1}},{{π,{-3,0,3,0,-3,0,3,0,-3,0,-8,0,-3,0,3,0,-3,0,3,0,-3,0,1},-1,22,1}},{{π,{-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,1},-1,26,1}},
{{π,{-3,0,-1,0,-2,0,3,0,1,0,3,0,-3,0,0,0,-3,0,3,0,1,0,3,0,-2,0,-1,0,-3,0,1},-1,30,1}},
{{π,{-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,1},-1,34,1}},
{{π,{-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,1},-1,38,1}},{{π,{-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,-16,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,1},-1,38,1}},
{{π,{-2,0,-1,0,-2,0,-3,0,1,0,2,0,-2,0,-1,0,-2,0,2,0,0,0,2,0,-2,0,-1,0,-2,0,2,0,1,0,-3,0,-2,0,-1,0,-2,0,1},-1,42,1}},
{{π,{-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,4,0,-4,0,1},-1,46,1}},{{π,{-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,-20,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,3,0,-3,0,1},-1,46,1}},
{{π,{-1,0,3,0,0,0,0,-21,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-21,0,0,0,0,0,0,0,-2,1},-1,48,1}}
};

(eqns=BasisSum[#,k]&/@Select[Flatten[l,1],GCD@@Flatten[Position[Most[#[[2]]],_?(#!=0&)]]==1&])//ColumnForm

N[ReleaseHold[eqns]]

(2^k)

ll={{{2,4},8,{2.2904570000002877` Second,{{π,{-4,0,0,2,1,1,0,0,1},16,8,1},{π,{0,-8,-4,-4,0,0,1,0,2},16,8,1}}}},
{{2,8},16,{6.169275999999627` Second,{{π,{-64,0,0,32,16,16,0,0,-4,0,0,2,1,1,0,0,16},256,16,1},{π,{0,-128,-64,-64,0,0,16,0,0,-8,-4,-4,0,0,1,0,32},256,16,1}}}},
{{2,12},24,{11.958969999999681` Second,{{π,{0,-256,-192,256,0,96,0,96,0,-16,0,4,0,0,3,6,0,2,0,1,0,0,0,0,32},4096,24,1},{π,{-256,-256,384,256,64,0,0,-96,-64,-16,0,-8,4,0,-6,-6,-1,-1,0,1,1,0,0,0,64},4096,24,1},{π,{0,-256,-64,0,-128,-352,-64,-288,-128,-80,0,-20,0,16,1,-6,2,0,1,-1,2,0,0,0,96},4096,24,1},{π,{-256,0,-320,-256,192,224,64,192,64,64,0,28,4,0,5,0,-3,-1,-1,0,-1,1,0,0,96},4096,24,1},{π,{512,-256,64,-512,0,-32,64,96,64,48,0,-12,-8,-16,-1,-6,0,-2,-1,-1,-1,0,0,0,-96},4096,24,1}}}}};

l=Flatten[#[[-1,-1]]&/@ll,1]

l={{π,{-4,0,0,2,1,1,0,0,1},16,8,1},{π,{0,-8,-4,-4,0,0,1,0,2},16,8,1},{π,{-64,0,0,32,16,16,0,0,-4,0,0,2,1,1,0,0,16},256,16,1},{π,{0,-128,-64,-64,0,0,16,0,0,-8,-4,-4,0,0,1,0,32},256,16,1},{π,{0,-256,-192,256,0,96,0,96,0,-16,0,4,0,0,3,6,0,2,0,1,0,0,0,0,32},4096,24,1},{π,{-256,-256,384,256,64,0,0,-96,-64,-16,0,-8,4,0,-6,-6,-1,-1,0,1,1,0,0,0,64},4096,24,1},{π,{0,-256,-64,0,-128,-352,-64,-288,-128,-80,0,-20,0,16,1,-6,2,0,1,-1,2,0,0,0,96},4096,24,1},{π,{-256,0,-320,-256,192,224,64,192,64,64,0,28,4,0,5,0,-3,-1,-1,0,-1,1,0,0,96},4096,24,1},{π,{512,-256,64,-512,0,-32,64,96,64,48,0,-12,-8,-16,-1,-6,0,-2,-1,-1,-1,0,0,0,-96},4096,24,1},
{π,{16384,0,0,-8192,-4096,-4096,0,0,1024,0,0,-512,-256,-256,0,0,64,0,0,-32,-16,-16,0,0,4,0,0,-2,-1,-1,0,0,-4096},65536,32,1},{π,{0,-32768,-16384,-16384,0,0,4096,0,0,-2048,-1024,-1024,0,0,256,0,0,-128,-64,-64,0,0,16,0,0,-8,-4,-4,0,0,1,0,8192},65536,32,1},
{π,{-262144,0,0,131072,65536,65536,0,0,-16384,0,0,8192,4096,4096,0,0,-1024,0,0,512,256,256,0,0,-64,0,0,32,16,16,0,0,-4,0,0,2,1,1,0,0,65536},1048576,40,1},{π,{0,0,-262144,-262144,-163840,-131072,65536,0,0,-40960,-16384,-16384,0,-8192,-1024,0,0,0,-1024,-1024,0,-512,256,0,160,0,-64,-64,0,8,16,0,0,0,1,-4,0,-2,1,0,65536},1048576,40,1},{π,{0,262144,262144,-131072,-163840,196608,-65536,163840,0,57344,16384,-8192,0,12288,11264,10240,0,1024,1024,2048,0,768,-256,640,160,64,64,-32,0,8,-16,40,0,4,-11,-2,0,3,-1,0,-65536},1048576,40,1},{π,{262144,262144,0,262144,98304,-262144,0,-163840,16384,-24576,0,16384,-4096,-16384,-15360,-10240,1024,1024,0,-1536,-256,-1024,0,-640,-96,64,0,64,-16,-24,0,-40,4,4,15,4,-1,-4,0,0,-131072},1048576,40,1}};

(eqns=BasisSum[#,k]&/@l)//ColumnForm

N[ReleaseHold[eqns]]

FindBasis[Pi,2^20,40]//Timing

FindBasis[Pi,2^20,50]//Timing

FindBasis[Pi,2^20,100]//Timing

FindBasis[Pi,2^20,50]//Timing

FindBasis[Pi,2^30,60]//Timing

FindBasis[Pi,2^30,60]//Timing

π Hybrid

π==4 (∑_(k=0)^∞)((((-1))^k)/((4^k) (2 k+1)))-(1/64) (∑_(k=0)^∞)(((((-1))^k) )/(1024^k))((8/(4 k+2))+(1/(4 k+3))+(32/(4 k+1)))

4Sum[((((-1))^k)/((4^k)(2k+1))),{k,0,∞}]-(1/64)Sum[((((-1))^k)/(1024^k))((32/(4k+1))+(8/(4k+2))+(1/(4k+3))),{k,0,∞}]

FullSimplify[FunctionExpand[%]]

(fb=FindBasis[Pi,-1024,128])//Timing

b=BasisSum[#,k]&/@fb

π==(1/4) (∑_(k=0)^∞)((((-1))^k)/(64^k)) ((4/(4 k+2))+(1/(4 k+3))+(8/(4 k+1)))+(1/64) (∑_(k=0)^∞)((((-1))^k)/(1024^k)) ((8/(4 k+2))+(1/(4 k+3))+(32/(4 k+1)))

(1/4)Sum[((((-1))^k)/(64^k))((8/(4k+1))+(4/(4k+2))+(1/(4k+3))),{k,0,∞}]+(1/64)Sum[((((-1))^k)/(1024^k))((32/(4k+1))+(8/(4k+2))+(1/(4k+3))),{k,0,∞}]

FullSimplify[FunctionExpand[%]]

(fb=FindBasis[π,-64,4])//Timing

(fb=FindBasis[π,-1024,64])//Timing

b=BasisSum[#,k]&/@fb

(Sqrt[3])π

(Sqrt[3]) π==(1/4) (∑_(k=0)^∞)(1/(64^k)) ((20/(1+6 k))+(6/(2+6 k))-(1/(3+6 k))-(3/(4+6 k))-(1/(5+6 k)))

(9/32)Sum[(1/(64^k))((16/(6k+1))+(8/(6k+2))-(2/(6k+4))-(1/(6k+5))),{k,0,∞}]

FullSimplify[%]

fb=FindBasis[(Sqrt[3])π,64,6]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

"Missed" solution

Total[{-8,-1}(#[[2]]&/@fb)]

mysoln=Flatten[{9{16,8,0,-2,-1,0},-32}]

N[mysoln.Append[v,Sqrt[3]Pi],1000]//InputForm

{(Sqrt[3]) π==(1/9) (∑_(k=0)^∞)(1/(729^k)) (-(54/(12 k+2))-(9/(12 k+4))-(12/(12 k+6))-(3/(12 k+7))-(2/(12 k+8))-(1/(12 k+10))+(81/(12 k+1))),(Sqrt[3]) π==(1/9) (∑_(k=0)^∞)(1/(729^k)) ((27/(12 k+2))-(162/(12 k+3))-(9/(12 k+4))+(27/(12 k+5))+(24/(12 k+6))-(3/(12 k+7))+(7/(12 k+8))+(6/(12 k+9))+(3/(12 k+10))-(1/(12 k+11))+(81/(12 k+1))),(Sqrt[3]) π==(1/36) (∑_(k=0)^∞)(1/(729^k)) ((189/(12 k+2))+(45/(12 k+4))+(27/(12 k+5))+(24/(12 k+6))-(3/(12 k+7))+(1/(12 k+8))+(1/(12 k+10))-(1/(12 k+11))+(81/(12 k+1)))}

(1/9)Sum[(1/(729^k))Plus@@MapIndexed[((#1)/(12k+#2[[1]]))&,{81,-54,0,-9,0,-12,-3,-2,0,-1}],{k,0,∞}]

FullSimplify[%]

Developer`ZeroQ[%73-(Sqrt[3])π]

fb=FindBasis[(Sqrt[3])π,729,12]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

N[%]

(π^2)

(((-1))^k)

l={{{(π^2),{-12,1},-1,1,2}},{{(π^2),{0,-48,1},-1,2,2}},{{(π^2),{-13,13,-4,1},-1,3,2}},{{(π^2),{0,0,0,-192,1},-1,4,2}},{{(π^2),{-12,12,-12,12,-12,1},-1,5,2}},{{(π^2),{0,-52,0,52,0,-16,1},-1,6,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,1},-1,7,2},{(π^2),{13,-13,13,-13,13,-13,-36,-1},-1,7,2}},{{(π^2),{0,0,0,0,0,0,0,-768,1},-1,8,2}},{{(π^2),{-13,13,-4,13,-13,4,-13,13,-4,1},-1,9,2}},{{(π^2),{0,-50,0,50,0,-50,0,50,0,0,1},-1,10,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,11,2},{(π^2),{13,-13,13,-13,13,-13,13,-13,13,-13,-108,-1},-1,11,2}},{{(π^2),{3,1,-25,-206,-3,-10,-2,206,-15,1,2,-80,1},-1,12,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,13,2},{(π^2),{-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,156,1},-1,13,2}},{{(π^2),{0,-49,0,49,0,-49,0,49,0,-49,0,49,0,0,1},-1,14,2}},{{(π^2),{-14,14,1,13,-1,0,-14,13,-8,1,-14,9,-13,13,-4,1},-1,15,2}},{{(π^2),{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3072,1},-1,16,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,17,2}},{{(π^2),{0,-52,0,53,0,-18,0,52,0,-53,0,9,0,-52,0,53,0,0,1},-1,18,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,19,2},{(π^2),{-25,25,-25,25,-25,25,-25,25,-25,25,-25,25,-25,25,-25,25,-25,25,336,2},-1,19,2}},{{(π^2),{-1,1,0,-199,26,-1,0,199,-1,-24,0,-199,1,-1,0,199,-1,1,0,-24,1},-1,20,2}},{{(π^2),{-13,13,-6,13,-13,6,-1,13,-6,13,-13,6,-13,1,-6,13,-13,6,-13,13,-4,1},-1,21,2}},{{(π^2),{0,-49,0,49,0,-49,0,49,0,-49,0,49,0,-49,0,49,0,-49,0,49,0,72,1},-1,22,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,23,2},{(π^2),{13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,-516,-1},-1,23,2}},{{(π^2),{4,7,-27,-5,8,-68,-4,-843,-85,-7,-9,50,-8,5,-68,843,4,52,9,-5,44,-5,-4,-168,1},-1,24,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,25,2}},{{(π^2),{0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,1},-1,26,2}},{{(π^2),{-13,13,-4,13,-13,4,-13,13,-4,13,-13,4,-13,13,-4,13,-13,4,-13,13,-4,13,-13,4,-13,13,-4,1},-1,27,2}},{{(π^2),{-1,-1,1,-195,1,1,48,195,-1,-1,1,-195,1,50,-1,195,-1,-1,1,-195,-48,1,-1,195,-1,-1,1,-48,1},-1,28,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,29,2},{(π^2),{37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,-804,-3},-1,29,2}},{{(π^2),{1,-55,-2,55,-17,-8,0,54,-6,-6,0,8,0,-55,-8,55,-1,-7,-1,6,-6,-55,1,17,9,-55,-2,54,0,0,1},-1,30,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,31,2},{(π^2),{-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,924,3},-1,31,2}},{{(π^2),{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12288,1},-1,32,2}},{{(π^2),{-13,13,-6,13,-13,5,-13,14,3,14,1,5,-13,14,-5,13,-13,5,-14,14,-6,-1,-13,-4,-14,13,4,13,-13,-3,-14,14,4,1},-1,33,2}},{{(π^2),{0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,1},-1,34,2}},{{(π^2),{-13,13,-13,13,4,13,4,13,-13,-4,-13,13,-13,-4,4,13,-13,13,-13,-4,4,13,-13,13,4,13,-13,-4,-13,-4,-13,13,-13,13,-12,1},-1,35,2}},{{(π^2),{-3,-1,27,-210,-3,12,3,207,1,-1,-3,-49,3,1,27,210,3,-30,-3,-207,-28,1,3,76,-3,-1,-9,-210,-3,12,3,207,28,-1,-3,-40,1},-1,36,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,37,2},{(π^2),{-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,1356,1},-1,37,2}},{{(π^2),{0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,1},-1,38,2}},{{(π^2),{-13,13,-6,13,-13,6,-13,14,4,13,-14,5,7,13,-5,14,-13,-4,-14,13,-6,13,-13,-3,-13,-7,-5,13,-14,5,-13,14,4,14,-13,5,-14,13,4,1},-1,39,2}},{{(π^2),{5,6,-9,1,-116,-2,10,-803,-5,-156,9,-1,-9,2,215,803,5,6,-9,-24,9,-2,10,-803,-230,-6,9,-1,-9,52,-10,803,5,6,-259,1,9,-2,10,72,1},-1,40,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,41,2},{(π^2),{-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,1644,3},-1,41,2}},{{(π^2),{-1,-55,8,53,-1,0,4,53,-1,-52,-1,16,0,-6,9,55,0,2,1,55,10,-53,2,15,1,-53,17,6,-1,-27,0,53,9,-55,4,16,-2,-52,-2,53,0,0,1},-1,42,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,43,2},{(π^2),{-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,-37,37,1812,3},-1,43,2}},{{(π^2),{-1,-1,-1,-194,1,1,1,194,-1,-1,120,-194,1,1,1,194,-1,-1,-1,-194,1,122,1,194,-1,-1,-1,-194,1,1,1,194,120,-1,-1,-194,1,1,1,194,-1,-1,-1,48,1},-1,44,2}},{{(π^2),{-13,13,-8,13,-4,8,-13,13,0,4,-13,8,-13,13,-1,13,-13,0,-13,4,-8,13,-13,8,-4,13,0,13,-13,1,-13,13,-8,13,-4,0,-13,13,-8,4,-13,8,-13,13,-4,1},-1,45,2}},{{(π^2),{0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,0,48,0,-48,1},-1,46,2}},{{(π^2),{-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,12,-12,1},-1,47,2},{(π^2),{73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,73,-73,-2136,-6},-1,47,2}},{{(π^2),{17,-6,-136,-25,-16,78,23,-15,-185,-12,29,220,26,24,133,-3320,-17,-210,-17,25,-191,-24,-23,150,32,12,259,5,-26,84,11,3320,-244,-6,17,20,-16,24,-211,-15,-32,-228,29,-5,125,24,-11,-1088,1},-1,48,2}}};

BasisSum[#,k]&/@Select[Flatten[l,1],GCD@@Flatten[Position[Most[#[[2]]],_?(#!=0&)]]==1&]//ColumnForm

(2^k)

l={{(π^2),{-16,16,8,16,4,4,-2,0,1},16,8,2},{(π^2),{0,-64,0,64,0,32,0,64,0,16,0,16,0,-8,0,0,1},16,16,2},{(π^2),{0,0,-144,0,0,144,0,0,72,0,0,144,0,0,36,0,0,36,0,0,-18,0,0,0,1},16,24,2},{(π^2),{0,0,0,-256,0,0,0,256,0,0,0,128,0,0,0,256,0,0,0,64,0,0,0,64,0,0,0,-32,0,0,0,0,1},16,32,2},{(π^2),{0,0,0,0,-400,0,0,0,0,400,0,0,0,0,200,0,0,0,0,400,0,0,0,0,100,0,0,0,0,100,0,0,0,0,-50,0,0,0,0,0,1},16,40,2},{(π^2),{144,-216,-72,-54,9,0,-8},64,6,2},{(π^2),{0,-144,0,216,0,72,0,54,0,-9,0,0,2},64,12,2},{(π^2),{0,0,-1296,0,0,1944,0,0,648,0,0,486,0,0,-81,0,0,0,8},64,18,2},{(π^2),{0,0,0,-288,0,0,0,432,0,0,0,144,0,0,0,108,0,0,0,-18,0,0,0,0,1},64,24,2},{(π^2),{0,0,0,0,-3600,0,0,0,0,5400,0,0,0,0,1800,0,0,0,0,1350,0,0,0,0,-225,0,0,0,0,0,8},64,30,2},{(π^2),{0,0,0,0,0,-1296,0,0,0,0,0,1944,0,0,0,0,0,648,0,0,0,0,0,486,0,0,0,0,0,-81,0,0,0,0,0,0,2},64,36,2},{(π^2),{128,-128,-64,-128,-32,-32,16,0,8,-8,-4,-8,-2,-2,1,0,-8},256,16,2},{(π^2),{0,-128,0,128,0,64,0,128,0,32,0,32,0,-16,0,0,0,-8,0,8,0,4,0,8,0,2,0,2,0,-1,0,0,2},256,32,2},{(π^2),{-9216,13824,4608,3456,-576,0,-144,216,72,54,-9,0,512},4096,12,2},{(π^2),{2048,-2048,-1024,-2048,-512,-512,256,0,128,-128,-64,-128,-32,-32,16,0,8,-8,-4,-8,-2,-2,1,0,-128},4096,24,2},{(π^2),{0,-5120,1536,6656,0,1792,0,1152,-192,-320,0,-64,0,-80,-24,72,0,28,0,26,3,-5,0,0,64},4096,24,2},{(π^2),{0,0,82944,0,0,-124416,0,0,-41472,0,0,-31104,0,0,5184,0,0,0,0,0,1296,0,0,-1944,0,0,-648,0,0,-486,0,0,81,0,0,0,-512},4096,36,2},{(π^2),{32768,-32768,-16384,-32768,-8192,-8192,4096,0,2048,-2048,-1024,-2048,-512,-512,256,0,128,-128,-64,-128,-32,-32,16,0,8,-8,-4,-8,-2,-2,1,0,-2048},65536,32,2},{(π^2),{589824,-884736,-294912,-221184,36864,0,9216,-13824,-4608,-3456,576,0,144,-216,-72,-54,9,0,-32768},262144,18,2},{(π^2),{0,-589824,0,884736,0,294912,0,221184,0,-36864,0,0,0,-9216,0,13824,0,4608,0,3456,0,-576,0,0,0,-144,0,216,0,72,0,54,0,-9,0,0,8192},262144,36,2},{(π^2),{-524288,524288,262144,524288,131072,131072,-65536,0,-32768,32768,16384,32768,8192,8192,-4096,0,-2048,2048,1024,2048,512,512,-256,0,-128,128,64,128,32,32,-16,0,-8,8,4,8,2,2,-1,0,32768},1048576,40,2},{(π^2),{524288,-1835008,-262144,1441792,1507328,-458752,65536,-819200,32768,-114688,-16384,90112,-8192,-28672,-47104,-51200,2048,-7168,-1024,-7168,-512,-1792,256,-3200,-1472,-448,-64,352,-32,-112,16,-200,8,-28,46,22,-2,-7,1,0,-16384},1048576,40,2}};

BasisSum[#,k]&/@Select[l,GCD@@Flatten[Position[Most[#[[2]]],_?(#!=0&)]]==1&]//ColumnForm

(3^k)

ll={{{3,6},12,{3.4200000000000728` Second,{{(π^2),{486,-810,0,-162,-54,-144,-18,-18,0,-10,2,0,-27},729,12,2}}}},
{{3,6},24,{15.049999999999272` Second,{{(π^2),{0,-1944,0,3240,0,0,0,648,0,216,0,576,0,72,0,72,0,0,0,40,0,-8,0,0,27},729,24,2}}}},
{{3,6},36,{43.54999999999927` Second,{{(π^2),{0,0,486,0,0,-810,0,0,0,0,0,-162,0,0,-54,0,0,-144,0,0,-18,0,0,-18,0,0,0,0,0,-10,0,0,2,0,0,0,-3},729,36,2}}}},
{{3,12},24,{14.719999999999345` Second,{{(π^2),{-354294,590490,0,118098,39366,104976,13122,13122,0,7290,-1458,0,-486,810,0,162,54,144,18,18,0,10,-2,0,19683},531441,24,2}}}}};

l=Flatten[#[[-1,-1]]&/@ll,1]

BasisSum[#,k]&/@Select[l,GCD@@Flatten[Position[Most[#[[2]]],_?(#!=0&)]]==1&]//ColumnForm

Log[2] π

{π log(2)==(1/256) (∑_(k=0)^∞)(1/(4096^k)) (-(4096/(((24 k+2))^2))-(3584/(((24 k+3))^2))-(512/(((24 k+5))^2))-(1280/(((24 k+6))^2))-(256/(((24 k+7))^2))-(448/(((24 k+9))^2))-(256/(((24 k+10))^2))+(64/(((24 k+11))^2))-(32/(((24 k+13))^2))+(64/(((24 k+14))^2))+(56/(((24 k+15))^2))+(8/(((24 k+17))^2))+(20/(((24 k+18))^2))+(4/(((24 k+19))^2))+(7/(((24 k+21))^2))+(4/(((24 k+22))^2))-(1/(((24 k+23))^2))+(2048/(((24 k+1))^2))),π log(2)==(1/256) (∑_(k=0)^∞)(1/(4096^k)) (-(2560/(((24 k+3))^2))-(11776/(((24 k+4))^2))-(5376/(((24 k+6))^2))-(512/(((24 k+7))^2))-(3456/(((24 k+8))^2))-(576/(((24 k+9))^2))+(448/(((24 k+10))^2))+(128/(((24 k+11))^2))+(128/(((24 k+12))^2))+(240/(((24 k+14))^2))+(40/(((24 k+15))^2))-(216/(((24 k+16))^2))-(44/(((24 k+18))^2))+(8/(((24 k+19))^2))-(46/(((24 k+20))^2))+(9/(((24 k+21))^2))+(15/(((24 k+22))^2))-(2/(((24 k+23))^2))+(7168/(((24 k+2))^2)))}

(1/256)Sum[(1/(4096^k))Plus@@MapIndexed[((#1)/(((24k+#2[[1]]))^2))&,{4096,-8192,-26112,15360,-1024,9984,0,11520,2368,-512,0,768,-64,0,408,720,16,196,0,60,-37}],{k,0,∞}]

fb=FindBasis[π Log[2],4096,24,2]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

N[%]

Catalan

(((-1))^k)

l={{-1,1,{0.02 Second,{}}},{-1,2,{0.04 Second,{{Catalan,{-1,0,1},-1,2,2}}}},{-1,3,{2.59 Second,{}}},{-1,4,{0.73 Second,{{Catalan,{0,-4,0,0,1},-1,4,2}}}},{-1,5,{4.45 Second,{}}},{-1,6,{2.3 Second,{{Catalan,{-1,0,1,0,-1,0,1},-1,6,2},{Catalan,{-2,0,-7,0,-2,0,3},-1,6,2}}}},{-1,7,{6.46 Second,{}}},{-1,8,{1.88 Second,{{Catalan,{0,0,0,-16,0,0,0,0,1},-1,8,2}}}},{-1,9,{6.64 Second,{}}},{-1,10,{4.95 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1},-1,10,2},{Catalan,{4,0,-4,0,-21,0,-4,0,4,0,-3},-1,10,2}}}},{-1,11,{11.68 Second,{}}},{-1,12,{4.99 Second,{{Catalan,{0,-4,0,0,0,4,0,0,0,-4,0,0,1},-1,12,2},{Catalan,{0,-3,0,0,0,-6,0,0,0,-3,0,0,1},-1,12,2}}}},{-1,13,{14.34 Second,{}}},{-1,14,{8.08 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,14,2},{Catalan,{-5,0,5,0,-5,0,-44,0,-5,0,5,0,-5,0,6},-1,14,2}}}},{-1,15,{9.38 Second,{}}},{-1,16,{6.07 Second,{{Catalan,{0,0,0,0,0,0,0,-64,0,0,0,0,0,0,0,0,1},-1,16,2}}}},{-1,17,{20.63 Second,{}}},{-1,18,{9.14 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,18,2},{Catalan,{0,0,-8,0,-1,0,0,0,-1,0,1,0,0,0,1,0,-1,0,1},-1,18,2},{Catalan,{-1,0,1,0,0,0,1,0,-1,0,0,0,-1,0,-8,0,0,0,1},-1,18,2},{Catalan,{1,0,0,0,1,0,-1,0,-9,0,-1,0,1,0,0,0,1,0,-1},-1,18,2}}}},{-1,19,{24.56 Second,{}}},{-1,20,{12.74 Second,{{Catalan,{0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,1},-1,20,2},{Catalan,{0,5,0,0,0,-5,0,0,0,-20,0,0,0,-5,0,0,0,5,0,0,-1},-1,20,2}}}},{-1,21,{15.53 Second,{}}},{-1,22,{18.29 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,22,2},{Catalan,{-10,0,10,0,-10,0,10,0,-10,0,-111,0,-10,0,10,0,-10,0,10,0,-10,0,11},-1,22,2},{Catalan,{-2,-1,2,1,-2,-1,2,1,-2,-1,2,1,-2,-1,2,1,-2,-1,2,1,-2,120,2},-1,22,2}}}},{-1,23,{32.73 Second,{}}},{-1,24,{13.94 Second,{{Catalan,{0,0,0,-14,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,-14,0,0,0,0,1},-1,24,2}}}},{-1,25,{28.24 Second,{}}},{-1,26,{26.31 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,26,2},{Catalan,{8,0,-8,0,8,0,-8,0,8,0,-8,0,-161,0,-8,0,8,0,-8,0,8,0,-8,0,8,0,-7},-1,26,2},{Catalan,{-2,1,2,-1,-2,1,2,-1,-2,1,2,-1,-2,1,2,-1,-2,1,2,-1,-2,1,2,-1,-2,-168,2},-1,26,2}}}},{-1,27,{27.29 Second,{}}},{-1,28,{26.38 Second,{{Catalan,{0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,1},-1,28,2},{Catalan,{0,-7,0,0,0,7,0,0,0,-7,0,0,0,-42,0,0,0,-7,0,0,0,7,0,0,0,-7,0,0,2},-1,28,2}}}},{-1,29,{47.2 Second,{}}},{-1,30,{20.22 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,30,2},{Catalan,{-1,0,0,0,2,0,1,0,0,0,1,0,-1,0,-5,0,-1,0,1,0,0,0,1,0,2,0,0,0,-1,0,1},-1,30,2},{Catalan,{0,0,-8,0,-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1},-1,30,2},{Catalan,{1,0,0,0,1,0,-1,0,-8,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1},-1,30,2},{Catalan,{-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,0,0,1,0,8,0,1,0,-1,0,0,0,-1,0,1},-1,30,2},{Catalan,{-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,-8,0,0,0,1},-1,30,2}}}},{-1,31,{52.49 Second,{}}},{-1,32,{30.25 Second,{{Catalan,{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-256,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1},-1,32,2}}}},{-1,33,{39.14 Second,{}}},{-1,34,{56.63 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,34,2},{Catalan,{16,0,-16,0,16,0,-16,0,16,0,-16,0,16,0,-16,0,-273,0,-16,0,16,0,-16,0,16,0,-16,0,16,0,-16,0,16,0,-15},-1,34,2}}}},{-1,35,{47.16 Second,{}}},{-1,36,{42.72 Second,{{Catalan,{0,-3,0,0,0,-5,0,0,0,-3,0,0,0,3,0,0,0,-4,0,0,0,3,0,0,0,-3,0,0,0,-5,0,0,0,-3,0,0,1},-1,36,2}}}},{-1,37,{70.82 Second,{}}},{-1,38,{70.83 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,38,2},{Catalan,{-18,0,18,0,-18,0,18,0,-18,0,18,0,-18,0,18,0,-18,0,-343,0,-18,0,18,0,-18,0,18,0,-18,0,18,0,-18,0,18,0,-18,0,19},-1,38,2}}}},{-1,39,{56.95 Second,{}}},{-1,40,{62.37 Second,{{Catalan,{0,0,0,-17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,-17,0,0,0,0,1},-1,40,2}}}},{-1,41,{83.6 Second,{}}},{-1,42,{58.4 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,42,2},{Catalan,{0,0,-8,0,-1,0,1,0,-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,42,2},{Catalan,{1,0,0,0,1,0,-1,0,-8,0,-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1,0,-1},-1,42,2},{Catalan,{-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,-8,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1},-1,42,2},{Catalan,{-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,0,0,-1,0,-8,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1},-1,42,2},{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,0,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,0,0,-1,0,1,0,-1,0,1,0,-1,0,-8,0,0,0,1},-1,42,2}}}},{-1,43,{91.68 Second,{}}},{-1,44,{86.75 Second,{{Catalan,{0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,0,4,0,0,0,-4,0,0,1},-1,44,2},{Catalan,{0,-11,0,0,0,11,0,0,0,-11,0,0,0,11,0,0,0,-11,0,0,0,-110,0,0,0,-11,0,0,0,11,0,0,0,-11,0,0,0,11,0,0,0,-11,0,0,3},-1,44,2},{Catalan,{0,4,1,0,0,-4,-1,0,0,4,1,0,0,-4,-1,0,0,4,1,0,0,-4,-1,0,0,4,1,0,0,-4,-1,0,-121,4,1,0,0,-4,-1,0,0,4,1,0,-1},-1,44,2}}}},{-1,45,{65.27 Second,{}}},{-1,46,{99.04 Second,{{Catalan,{-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1},-1,46,2},{Catalan,{-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,-507,0,-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,22,0,-22,0,23},-1,46,2}}}},{-1,47,{106.81 Second,{}}},{-1,48,{70.06 Second,{{Catalan,{0,0,0,0,0,0,0,-57,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-57,0,0,0,0,0,0,0,0,1},-1,48,2}}}}};

Length[#[[3,2]]]&/@l

Last/@Partition[{0,1,0,1,0,2,0,1,0,2,0,2,0,2,0,1,0,4,0,2,0,3,0,1,0,3,0,2,0,6,0,1,0,2,0,1,0,2,0,1,0,6,0,3,0,2,0,1},2]

FindBasis[Catalan,-8,12,BasisPower->2]

FindBasis[Catalan,-16,12,BasisPower->2]

FindBasis[Catalan,-32,12,BasisPower->2]

-Integrate[3 (2-4 x+x^2)/(4-4 x+2 x^2-2 x^3+x^4) Log[x],{x,0,1}]//FullSimplify

N[%]

Catalan//N

FindBasis[Catalan,-64,12,BasisPower->2]

BasisSum[#,k]&/@%//TraditionalForm

{C==(3/64) (∑_(k=0)^∞)(1/(((-64))^k)) ((32/(((12 k+1))^2))-(32/(((12 k+2))^2))-(32/(((12 k+3))^2))-(8/(((12 k+5))^2))-(16/(((12 k+6))^2))-(4/(((12 k+7))^2))-(4/(((12 k+9))^2))-(2/(((12 k+10))^2))+(1/(((12 k+11))^2)))}

sums2=Map[BasisSum[#,k]&,#[[-1,-1]]&/@l,{2}]

C==(∑_(k=0)^∞)((((-1))^k)/(((2 k+1))^2))

Sum[((((-1))^k)/(((2k+1))^2)),{k,0,∞}]

fb=FindBasis[Catalan,-1,2,2]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

{C==(1/1024)((∑_(k=0)^∞)(1/(4096^k)) ((1024/(((24 k+2))^2))-(512/(((24 k+3))^2))-(3072/(((24 k+4))^2))-(256/(((24 k+5))^2))-(2048/(((24 k+6))^2))-(256/(((24 k+7))^2))-(1152/(((24 k+8))^2))-(320/(((24 k+9))^2))+(64/(((24 k+10))^2))+(64/(((24 k+11))^2))-(16/(((24 k+13))^2))+(64/(((24 k+14))^2))+(8/(((24 k+15))^2))-(72/(((24 k+16))^2))+(4/(((24 k+17))^2))-(8/(((24 k+18))^2))+(4/(((24 k+19))^2))-(12/(((24 k+20))^2))+(5/(((24 k+21))^2))+(4/(((24 k+22))^2))-(1/(((24 k+23))^2))+(1024/(((24 k+1))^2)))),C==(1/3072)((∑_(k=0)^∞)(1/(4096^k)) (-(8192/(((24 k+2))^2))-(2560/(((24 k+3))^2))+(2560/(((24 k+4))^2))-(1280/(((24 k+5))^2))-(2048/(((24 k+6))^2))-(512/(((24 k+7))^2))-(832/(((24 k+9))^2))-(512/(((24 k+10))^2))+(128/(((24 k+11))^2))-(128/(((24 k+12))^2))-(80/(((24 k+13))^2))+(16/(((24 k+14))^2))+(40/(((24 k+15))^2))+(20/(((24 k+17))^2))+(40/(((24 k+18))^2))+(8/(((24 k+19))^2))+(10/(((24 k+20))^2))+(13/(((24 k+21))^2))+(1/(((24 k+22))^2))-(2/(((24 k+23))^2))+(5120/(((24 k+1))^2))))}

(1/1024)Sum[(1/(4096^k))Plus@@MapIndexed[((#1)/(((24k+#2[[1]]))^2))&,{3072,-3072,-23040,12288,-768,9216,0,10368,2496,-192,0,768,-48,0,360,648,12,168,0,48,-39}],{k,0,∞}]

{Catalan==(1/1024)((∑_(k=0)^∞)(1/(4096^k)) ((1024/(((1+24 k))^2))+(1024/(((2+24 k))^2))-(512/(((3+24 k))^2))-(3072/(((4+24 k))^2))-(256/(((5+24 k))^2))-(2048/(((6+24 k))^2))-(256/(((7+24 k))^2))-(1152/(((8+24 k))^2))-(320/(((9+24 k))^2))+(64/(((10+24 k))^2))+(64/(((11+24 k))^2))-(16/(((13+24 k))^2))+(64/(((14+24 k))^2))+(8/(((15+24 k))^2))-(72/(((16+24 k))^2))+(4/(((17+24 k))^2))-(8/(((18+24 k))^2))+(4/(((19+24 k))^2))-(12/(((20+24 k))^2))+(5/(((21+24 k))^2))+(4/(((22+24 k))^2))-(1/(((23+24 k))^2)))),
Catalan==(1/3072)((∑_(k=0)^∞)(1/(4096^k)) ((5120/(((1+24 k))^2))-(8192/(((2+24 k))^2))-(2560/(((3+24 k))^2))+(2560/(((4+24 k))^2))-(1280/(((5+24 k))^2))-(2048/(((6+24 k))^2))-(512/(((7+24 k))^2))-(832/(((9+24 k))^2))-(512/(((10+24 k))^2))+(128/(((11+24 k))^2))-(128/(((12+24 k))^2))-(80/(((13+24 k))^2))+(16/(((14+24 k))^2))+(40/(((15+24 k))^2))+(20/(((17+24 k))^2))+(40/(((18+24 k))^2))+(8/(((19+24 k))^2))+(10/(((20+24 k))^2))+(13/(((21+24 k))^2))+(1/(((22+24 k))^2))-(2/(((23+24 k))^2))))}

fb=FindBasis[Catalan,4096,24,BasisPower->2]

b=BasisSum[#,k]&/@fb

FullSimplify[ReleaseHold[b]]

N[%]

BasisSum[#,k]&/@{
{Catalan,{1024,1024,-512,-3072,-256,-2048,-256,-1152,-320,64,64,0,-16,64,8,-72,4,-8,4,-12,5,4,-1,0,-1024},4096,24,2},{Catalan,{5120,-8192,-2560,2560,-1280,-2048,-512,0,-832,-512,128,-128,-80,16,40,0,20,40,8,10,13,1,-2,0,-3072},4096,24,2}}

N[ReleaseHold[%],100]

Dirichlet L-Series

(Subscript[L, -7])(2)

Sum[(1/(((7n+1))^2))+(1/(((7n+2))^2))-(1/(((7n+3))^2))+(1/(((7n+4))^2))-(1/(((7n+5))^2))-(1/(((7n+6))^2)),{n,0,∞}]//FunctionExpand

fb=FindBasis[LSeries[-7,2],1,7,2]

b=BasisSum[#,k]&/@%

FullSimplify[ReleaseHold[b]]

Zeta

(((-1))^k)

l={{-1,1,{0.539062 Second,{{Zeta[3],{4,-3},-1,1,3}}}},{-1,2,{0.246094 Second,{{Zeta[3],{0,-32,3},-1,2,3}}}},{-1,3,{0.100586 Second,{{Zeta[3],{4,-4,4,-3},-1,3,3},{Zeta[3],{3,-3,-6,-2},-1,3,3}}}},{-1,4,{1.02051 Second,{{Zeta[3],{0,0,0,-256,3},-1,4,3}}}},{-1,5,{6.64941 Second,{{Zeta[3],{4,-4,4,-4,4,-3},-1,5,3},{Zeta[3],{-21,21,-21,21,104,15},-1,5,3}}}},{-1,6,{3.28613 Second,{{Zeta[3],{0,-11,0,11,0,-2,1},-1,6,3}}}},{-1,7,{10.2393 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-3},-1,7,3},{Zeta[3],{41,-41,41,-41,41,-41,-302,-30},-1,7,3}}}},{-1,8,{2.69043 Second,{{Zeta[3],{0,0,0,0,0,0,0,2048,-3},-1,8,3}}}},{-1,9,{10.6152 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,3},-1,9,3},{Zeta[3],{3,-3,-6,-3,3,6,3,-3,-6,-2},-1,9,3},{Zeta[3],{2,-1,-17,-2,1,-10,2,-1,10,-1},-1,9,3},{Zeta[3],{-1,2,-10,1,-2,-17,-1,2,-10,1},-1,9,3}}}},{-1,10,{7.81348 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,-3},-1,10,3},{Zeta[3],{0,33,0,-33,0,33,0,-33,0,-92,-3},-1,10,3}}}},{-1,11,{17.9824 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,11,3},{Zeta[3],{-133,133,-133,133,-133,133,-133,133,-133,133,1198,99},-1,11,3}}}},{-1,12,{8.20996 Second,{{Zeta[3],{0,0,0,-88,0,0,0,88,0,0,0,-16,1},-1,12,3}}}},{-1,13,{22.4004 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,13,3},{Zeta[3],{-165,165,-165,165,-165,165,-165,165,-165,165,-165,165,2032,123},-1,13,3}}}},{-1,14,{12.8936 Second,{{Zeta[3],{0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,3},-1,14,3},{Zeta[3],{0,33,0,-33,0,33,0,-33,0,33,0,-33,0,-310,-3},-1,14,3}}}},{-1,15,{15.1982 Second,{{Zeta[3],{-3,3,5,3,2,-5,-3,3,5,-2,-3,-5,-3,3,0,2},-1,15,3},{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,15,3},{Zeta[3],{-2,3,-21,3,-3,-7,-3,3,6,3,-2,-6,-3,3,6,2},-1,15,3},{Zeta[3],{3,-2,-5,-3,-2,-22,2,-3,-5,2,3,6,3,-3,0,-2},-1,15,3},{Zeta[3],{3,-3,-5,-2,-2,5,3,-3,-6,2,3,-22,3,-2,0,-2},-1,15,3},{Zeta[3],{-2,1,3,2,62,-3,-2,1,3,64,-1,-3,-2,1,-10,1},-1,15,3}}}},{-1,16,{10.1641 Second,{{Zeta[3],{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16384,-3},-1,16,3}}}},{-1,17,{32.8467 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,17,3},{Zeta[3],{189,-189,189,-189,189,-189,189,-189,189,-189,189,-189,189,-189,189,-189,-4724,-141},-1,17,3}}}},{-1,18,{14.5293 Second,{{Zeta[3],{0,-11,0,11,0,-2,0,11,0,-11,0,2,0,-11,0,11,0,-2,1},-1,18,3}}}},{-1,19,{38.8984 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,19,3},{Zeta[3],{-129,129,-129,129,-129,129,-129,129,-129,129,-129,129,-129,129,-129,129,-129,129,6730,96},-1,19,3}}}},{-1,20,{21.4814 Second,{{Zeta[3],{0,0,0,-258,0,0,0,258,0,0,0,-258,0,0,0,258,0,0,0,-8,3},-1,20,3}}}},{-1,21,{25.8662 Second,{{Zeta[3],{3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-2},-1,21,3},{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,21,3},{Zeta[3],{3,-2,-6,-3,3,-21,3,-3,-7,-3,3,6,3,-3,-6,-2,3,6,3,-3,-6,-2},-1,21,3},{Zeta[3],{-3,3,7,3,-3,-6,-3,3,-21,2,-3,-6,-3,3,6,3,-2,-6,-3,3,6,2},-1,21,3},{Zeta[3],{3,-3,-6,-2,3,6,3,-3,-6,-3,2,-21,3,-3,-6,-3,3,7,3,-3,-6,-2},-1,21,3},{Zeta[3],{-3,3,6,3,-3,-6,-2,3,6,3,-3,-6,-3,2,6,3,-3,-6,-3,3,-20,2},-1,21,3},{Zeta[3],{-3,2,-8,3,-2,8,182,2,-8,3,-2,8,-3,160,-8,3,-2,8,-3,2,-16,2},-1,21,3}}}},{-1,22,{30.3379 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,-3},-1,22,3},{Zeta[3],{1,-32,-1,32,1,-32,-1,32,1,-32,-1332,32,1,-32,-1,32,1,-32,-1,32,1,-32,3},-1,22,3},{Zeta[3],{0,-97,0,97,0,-97,0,97,0,-97,0,97,0,-97,0,97,0,-97,0,97,0,1234,9},-1,22,3}}}},{-1,23,{52.9141 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,23,3},{Zeta[3],{133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,133,-133,-12034,-99},-1,23,3}}}},{-1,24,{24.1543 Second,{{Zeta[3],{-2,0,53,1,-1,-1,-1,-709,29,0,1,-28,1,1,28,709,-2,27,-1,1,26,-1,-1,2,1},-1,24,3}}}},{-1,25,{49.0371 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,25,3},{Zeta[3],{5,-4,4,-4,-121,-5,4,-4,4,-4,5,-4,4,-4,4,-5,4,-4,4,-4,5,-4,4,-4,4,-3},-1,25,3},{Zeta[3],{-4,5,-4,4,-4,4,-5,4,-4,-121,-4,5,-4,4,-4,4,-5,4,-4,4,-4,5,-4,4,-4,3},-1,25,3},{Zeta[3],{4,-4,5,-4,4,-4,4,-5,4,-4,4,-4,5,-4,-121,-4,4,-5,4,-4,4,-4,5,-4,4,-3},-1,25,3},{Zeta[3],{-4,4,-4,5,-4,4,-4,4,-5,4,-4,4,-4,5,-4,4,-4,4,-5,-121,-4,4,-4,5,-4,3},-1,25,3},{Zeta[3],{4,-4,4,-4,5,-4,4,-4,4,-5,4,-4,4,-4,5,-4,4,-4,4,-5,4,-4,4,-4,-120,-3},-1,25,3}}}},{-1,26,{42.2275 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,-3},-1,26,3},{Zeta[3],{0,193,0,-193,0,193,0,-193,0,193,0,-193,0,193,0,-193,0,193,0,-193,0,193,0,-193,0,-2004,-18},-1,26,3}}}},{-1,27,{45.3135 Second,{{Zeta[3],{3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-2},-1,27,3},{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,27,3}}}},{-1,28,{44.7939 Second,{{Zeta[3],{0,0,0,257,0,0,0,-257,0,0,0,257,0,0,0,-257,0,0,0,257,0,0,0,-257,0,0,0,-86,-3},-1,28,3}}}},{-1,29,{78.0986 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,29,3},{Zeta[3],{961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,961,-961,-23428,-720},-1,29,3}}}},{-1,30,{35.3711 Second,{{Zeta[3],{0,-11,0,11,0,-3,0,11,0,-6,0,3,0,-11,0,11,0,-3,0,6,0,-11,0,3,0,-11,0,11,0,-8,1},-1,30,3}}}},{-1,31,{86.9092 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,31,3},{Zeta[3],{-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,-805,805,28986,603},-1,31,3}}}},{-1,32,{51.6328 Second,{{Zeta[3],{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,131072,-3},-1,32,3}}}},{-1,33,{67.8018 Second,{{Zeta[3],{3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-2},-1,33,3},{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,33,3},{Zeta[3],{7,-7,-1,-7,7,1,7,-7,-1,-7,-44,1,7,-7,-1,-7,7,1,7,-7,-1,44,7,1,7,-7,-1,-7,7,1,7,-7,-6,-5},-1,33,3},{Zeta[3],{14,-17,-19,-17,15,71,14,-17,34,-10,678,98,16,-13,-14,-14,15,17,16,-12,-21,654,15,69,12,-18,64,-14,15,-93,11,-11,48,-10},-1,33,3}}}},{-1,34,{95.1748 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,-3},-1,34,3},{Zeta[3],{1,-64,-1,64,1,-64,-1,64,1,-64,-1,64,1,-64,-1,64,-4912,-64,-1,64,1,-64,-1,64,1,-64,-1,64,1,-64,-1,64,1,-64,6},-1,34,3},{Zeta[3],{0,161,0,-161,0,161,0,-161,0,161,0,-161,0,161,0,-161,0,161,0,-161,0,161,0,-161,0,161,0,-161,0,161,0,-161,0,-4752,-15},-1,34,3}}}},{-1,35,{76.5605 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,35,3},{Zeta[3],{5,-4,4,-4,-117,-4,-7,-5,4,-8,4,-4,4,7,9,-4,4,-4,4,-8,-7,-5,4,-4,8,-4,4,7,5,-8,4,-4,4,-4,0,-3},-1,35,3},{Zeta[3],{8,-8,9,-8,12,-8,-3,-8,8,-13,8,-8,8,3,-113,-8,9,-8,8,-12,-3,-8,8,-9,12,-8,8,3,8,-12,9,-8,8,-8,4,-6},-1,35,3},{Zeta[3],{-4,4,-4,5,-8,4,7,4,-4,8,-5,4,-4,-7,-8,4,-4,5,-4,-117,7,4,-4,4,-9,4,-4,-7,-4,8,-4,5,-4,4,0,3},-1,35,3},{Zeta[3],{4,-4,4,-4,9,-4,-7,-4,4,-8,4,-5,4,7,8,-4,4,-4,5,-8,-7,-4,4,-4,-117,-5,4,7,4,-8,4,-4,5,-4,0,-3},-1,35,3},{Zeta[3],{-4,4,-4,4,-8,5,7,4,-4,8,-4,4,-5,-7,-8,4,-4,4,-4,9,7,4,-4,4,-8,4,-5,-7,-4,-117,-4,4,-4,5,0,3},-1,35,3},{Zeta[3],{8,-8,8,-9,-17,-8,77,-8,9,17,8,-8,8,-78,-17,-8,8,-8,9,17,77,-8,8,-9,-17,-8,8,266,9,17,8,-8,8,-9,2,-6},-1,35,3},{Zeta[3],{-12,12,-13,12,12,12,-78,13,-12,-12,-12,12,-13,78,12,12,-12,13,-12,-12,265,12,-13,12,12,12,-12,79,-12,-12,-12,12,-13,12,-36,9},-1,35,3}}}},{-1,36,{77.8965 Second,{{Zeta[3],{0,0,0,-88,0,0,0,88,0,0,0,-17,0,0,0,88,0,0,0,-88,0,0,0,17,0,0,0,-88,0,0,0,88,0,0,0,10,1},-1,36,3}}}},{-1,37,{118.47 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,37,3},{Zeta[3],{-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,-1057,1057,49596,792},-1,37,3}}}},{-1,38,{122.037 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,-3},-1,38,3},{Zeta[3],{1,-96,-1,96,1,-96,-1,96,1,-96,-1,96,1,-96,-1,96,1,-96,-6860,96,1,-96,-1,96,1,-96,-1,96,1,-96,-1,96,1,-96,-1,96,1,-96,9},-1,38,3},{Zeta[3],{0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,-321,0,321,0,6538,30},-1,38,3}}}},{-1,39,{97.9766 Second,{{Zeta[3],{-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,3,-3,-6,-3,3,6,2},-1,39,3},{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,39,3},{Zeta[3],{6,-5,-11,-6,6,-15,6,-5,-11,-5,6,12,-79,-6,-12,-6,6,-16,5,-6,-12,-6,5,-16,5,79,-11,-5,6,-16,6,-5,-11,-5,6,-15,6,-5,2,-4},-1,39,3},{Zeta[3],{-12,13,10,12,-8,-62,-10,12,12,14,-9,-9,1169,11,-73,12,-11,-67,-14,9,-45,10,-13,-39,-13,1027,-44,14,-11,-64,-9,13,-42,13,-9,-35,-10,12,62,8},-1,39,3}}}},{-1,40,{104.321 Second,{{Zeta[3],{-1,-3,2,1,126,-3,-1,2062,1,378,-2,-1,-1,3,-249,-2062,-1,-3,2,-124,1,-3,-1,2062,-124,3,-2,-1,-1,378,1,-2062,-1,-3,127,1,1,-3,-1,312,-3},-1,40,3}}}},{-1,41,{145.436 Second,{{Zeta[3],{-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,3},-1,41,3},{Zeta[3],{-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,-1465,1465,67456,1098},-1,41,3}}}},{-1,42,{101.802 Second,{{Zeta[3],{0,-11,0,11,0,-3,0,11,0,-11,0,3,0,2,0,11,0,-3,0,11,0,-11,0,3,0,-11,0,-2,0,-3,0,11,0,-11,0,3,0,-11,0,11,0,2,1},-1,42,3}}}},{-1,43,{155.612 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,43,3},{Zeta[3],{-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,-2025,2025,77482,1518},-1,43,3}}}},{-1,44,{151.556 Second,{{Zeta[3],{0,0,0,256,0,0,0,-256,0,0,0,256,0,0,0,-256,0,0,0,256,0,0,0,-256,0,0,0,256,0,0,0,-256,0,0,0,256,0,0,0,-256,0,0,0,256,-3},-1,44,3}}}},{-1,45,{112.805 Second,{{Zeta[3],{-3,3,5,3,3,-5,-3,3,0,-3,-3,-5,-3,3,-3,3,-3,0,-3,-3,5,3,-3,-5,3,3,0,3,-3,3,-3,3,5,3,3,0,-3,3,5,-3,-3,-5,-3,3,-4,2},-1,45,3},{Zeta[3],{-4,4,-4,4,-5,5,-4,4,0,0,-4,4,-4,4,3,4,-4,0,-4,5,-4,4,-4,5,0,4,0,4,-4,6,-4,4,-5,4,-5,0,-4,4,-4,0,-4,5,-4,4,-6,3},-1,45,3},{Zeta[3],{-4,3,15,3,82,-6,-2,3,-3,2,-3,-15,-2,3,4,3,-3,3,-4,43,-12,2,-3,-6,-1,3,-3,3,-3,-4,-3,3,6,3,-43,3,-3,3,-12,1,-3,-6,-2,3,4,2},-1,45,3},{Zeta[3],{2,-2,-17,-1,3,-1,1,-1,0,80,2,-10,1,-1,-5,-2,1,0,1,-4,10,-1,1,-1,45,-1,0,-1,2,5,2,-1,1,-1,3,0,1,-2,10,-45,1,-1,1,-1,2,-1},-1,45,3},{Zeta[3],{-1,1,-10,1,-3,1,-1,2,0,45,-1,10,-1,1,5,1,-2,0,-1,3,-10,1,-1,1,-45,2,0,1,-1,-5,-1,1,-1,1,-4,0,-1,1,-10,-80,-1,1,-1,2,-2,1},-1,45,3}}}},{-1,46,{170.945 Second,{{Zeta[3],{0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,0,-32,0,32,-3},-1,46,3},{Zeta[3],{1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-12168,-64,1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-1,-64,1,64,-6},-1,46,3},{Zeta[3],{0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,833,0,-833,0,-11334,-78},-1,46,3}}}},{-1,47,{179.329 Second,{{Zeta[3],{4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-4,4,-3},-1,47,3},{Zeta[3],{-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,-3477,3477,100346,2607},-1,47,3}}}},{-1,48,{124.507 Second,{{Zeta[3],{0,-4,-12,13,-19,129,-14,-2,327,-18,8,-358,-20,14,506,-5671,0,-563,12,-13,397,-21,14,56,-3,18,-89,7,20,472,7,5671,-216,-4,-12,202,-19,21,526,-2,3,-396,8,-7,169,14,-7,-10,1},-1,48,3}}}}};

Length[#[[3,2]]]&/@l

sums2=Map[BasisSum[#,k]&,#[[-1,-1]]&/@l,{2}]

{Zeta[3]==(1/672) (∑_(k=0)^∞)(1/(4096^k)) ((2048/(((1+24 k))^3))-(11264/(((2+24 k))^3))-(1024/(((3+24 k))^3))+(11776/(((4+24 k))^3))-(512/(((5+24 k))^3))+(4096/(((6+24 k))^3))+(256/(((7+24 k))^3))+(3456/(((8+24 k))^3))+(128/(((9+24 k))^3))-(704/(((10+24 k))^3))-(64/(((11+24 k))^3))-(128/(((12+24 k))^3))-(32/(((13+24 k))^3))-(176/(((14+24 k))^3))+(16/(((15+24 k))^3))+(216/(((16+24 k))^3))+(8/(((17+24 k))^3))+(64/(((18+24 k))^3))-(4/(((19+24 k))^3))+(46/(((20+24 k))^3))-(2/(((21+24 k))^3))-(11/(((22+24 k))^3))+(1/(((23+24 k))^3))),Zeta[3]==(9/224) (∑_(k=0)^∞)(1/(4096^k)) ((1024/(((2+24 k))^3))-(3072/(((3+24 k))^3))+(512/(((4+24 k))^3))+(1024/(((6+24 k))^3))+(1152/(((8+24 k))^3))+(384/(((9+24 k))^3))+(64/(((10+24 k))^3))+(128/(((12+24 k))^3))+(16/(((14+24 k))^3))+(48/(((15+24 k))^3))+(72/(((16+24 k))^3))+(16/(((18+24 k))^3))+(2/(((20+24 k))^3))-(6/(((21+24 k))^3))+(1/(((22+24 k))^3)))}

(fb=FindBasis[Zeta[3],2^12,24,3])//Timing

b=BasisSum[#,k]&/@fb

FullSimplify[ReleaseHold[b]]

N[%13,10^4]

ArcTan

{ArcTan[(4/5)]==(1/131072)((∑_(k=0)^∞)(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k)))),
ArcTan[(4/5)]==(1/131072)((∑_(k=0)^∞)(1/(1048576^k)) ((393216/(4+40 k))+(163840/(5+40 k))-(131072/(6+40 k))-(163840/(8+40 k))+(24576/(12+40 k))-(8192/(14+40 k))-(15360/(15+40 k))-(10240/(16+40 k))-(1024/(20+40 k))-(512/(22+40 k))-(640/(24+40 k))-(160/(25+40 k))+(96/(28+40 k))-(32/(30+40 k))-(40/(32+40 k))+(15/(35+40 k))+(6/(36+40 k))-(2/(38+40 k)))),
ArcTan[(4/5)]==(1/131072)((∑_(k=0)^∞)(1/(1048576^k)) ((262144/(1+40 k))-(262144/(3+40 k))-(65536/(5+40 k))-(327680/(6+40 k))+(65536/(7+40 k))-(163840/(8+40 k))+(16384/(9+40 k))-(40960/(10+40 k))-(16384/(11+40 k))-(4096/(13+40 k))-(20480/(14+40 k))-(16384/(15+40 k))-(10240/(16+40 k))+(1024/(17+40 k))-(1024/(19+40 k))-(2560/(20+40 k))-(256/(21+40 k))-(1280/(22+40 k))+(256/(23+40 k))-(640/(24+40 k))+(64/(25+40 k))-(64/(27+40 k))-(16/(29+40 k))-(40/(30+40 k))+(16/(31+40 k))-(40/(32+40 k))+(4/(33+40 k))+(16/(35+40 k))-(1/(37+40 k))-(5/(38+40 k))+(1/(39+40 k)))),
ArcTan[(4/5)]==(1/262144)((∑_(k=0)^∞)(1/(1048576^k)) ((262144/(3+40 k))+(262144/(4+40 k))+(131072/(6+40 k))-(65536/(7+40 k))+(81920/(10+40 k))+(16384/(11+40 k))+(16384/(12+40 k))+(8192/(14+40 k))-(4096/(15+40 k))+(1024/(19+40 k))+(1024/(20+40 k))+(512/(22+40 k))-(256/(23+40 k))+(64/(27+40 k))+(64/(28+40 k))-(48/(30+40 k))-(16/(31+40 k))+(4/(35+40 k))+(4/(36+40 k))+(2/(38+40 k))-(1/(39+40 k))))}

(2^16)

fb=FindBasis[ArcTan[4/5],2^16,40]

(2^20)

(fb=FindBasis[ArcTan[4/5],2^20,40])//Timing

b=BasisSum[#,k]&/@%

N[ReleaseHold[b]]

(1/131072)((∑_(k=0)^∞)(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k))))

sum1=(1/131072)Sum[(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k))),{k,0,∞}]

N[ArcTan[(4/5)],20]

N[sum1,20]

sum1//FunctionExpand//FullSimplify//Timing

(1/20) (4 π+ArcTan[(10747406201475600/8041209044472401)])-ArcTan[(4/5)]//FullSimplify

Subtract@@{(1/131072)Sum[(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k))),{k,0,∞}]//FullSimplify,ArcTan[(4/5)]}//FullSimplify//Timing

Subtract@@{(1/131072)Sum[(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k))),{k,0,∞}]//FunctionExpand//FullSimplify,ArcTan[(4/5)]}//FullSimplify//Timing

Subtract@@{(1/131072)Sum[(1/(1048576^k)) ((262144/(2+40 k))-(163840/(5+40 k))-(65536/(6+40 k))+(16384/(10+40 k))-(4096/(14+40 k))-(5120/(15+40 k))+(1024/(18+40 k))-(256/(22+40 k))+(160/(25+40 k))+(64/(26+40 k))-(16/(30+40 k))+(4/(34+40 k))+(5/(35+40 k))-(1/(38+40 k))),{k,0,∞}],ArcTan[(4/5)]}//FunctionExpand//FullSimplify//Timing

g[a_,b_]:=NSum[(1/(1048576^k))(1/(a k+b)),{k,0,∞}]

(1/131072)(262144*g[40,2]-163840*g[40,5]-65536*g[40,6]+16384*g[40,10]-4096*g[40,14]-5120*g[40,15]+1024*g[40,18]-256*g[40,22]+160*g[40,25]+64*g[40,26]-16*g[40,30]+4*g[40,34]+5*g[40,35]-g[40,38])

Wrong Plouffe expression from Aug 24, 2024:

g[a_,b_]:=NSum[(1/(1048576^k))(1/(b k+a)),{k,0,∞}]

(1/131072)(262144*g[40,2]-163840*g[40,5]-65536*g[40,6]+16384*g[40,10]-4096*g[40,14]-5120*g[40,15]+1024*g[40,18]-256*g[40,22]+160*g[40,25]+64*g[40,26]-16*g[40,30]+4*g[40,34]+5*g[40,35]-g[40,38])

Base ϕ

(π^2)==(∑_(k=0)^∞)(1/(Φ^(5k)))(((Φ^(-2))/(((5k+1))^2))-((Φ^(-1))/(((5k+2))^2))-((Φ^(-2))/(((5k+3))^2))+((Φ^(-5))/(((5k+4))^2))+((2(Φ^(-5)))/(((5k+5))^2)))

Do[Print[{3^n,k,FindBasis[sigma,3^n,k]//Timing}],{n,20},{k,24}]

sum=Sum[(((GoldenRatio^(-2))/(((5k+1))^2))-((GoldenRatio^(-1))/(((5k+2))^2))-((GoldenRatio^(-2))/(((5k+3))^2))+((GoldenRatio^(-5))/(((5k+4))^2))+((2(GoldenRatio^(-5)))/(((5k+5))^2)))(1/(GoldenRatio^(5k))),{k,0,∞}]

N[{sum,((π^2)/50)},50]

FullSimplify[sum]