Somos's Quadratic Recurrence Constant

Author

Eric W. Weisstein
August 5, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/SomossQuadraticRecurrenceConstant.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Recurrence

Solution

g[0]:=1
g[n_]:=n g[n-1]^2

g/@Range[10]

v5.2

RSolve[{g[n]==n g[n-1]^2,g[0]==1},g[n],n]

...wrong...

expr=-(2^(-1+n)) Derivative[0,1,0][LerchPhi][(1/2),0,1]+(2^(-2+n)) Derivative[0,1,0][LerchPhi][(1/2),0,2]+(1/2) Derivative[0,1,0][LerchPhi][(1/2),0,1+n]-(1/4) Derivative[0,1,0][LerchPhi][(1/2),0,2+n];

Series[expr,{n,∞,2}]

Exp[Table[expr,{n,0,4}]]//N

v6

RSolve[{g[n]==n g[n-1]^2,g[0]==1},g[n],n]

FullSimplify[%,n>0&&n∈Integers]

Table[(E^(-(2^n) Derivative[1,0][PolyLog][0,(1/2)]+(1/2) Derivative[0,1,0][LerchPhi][(1/2),0,1+n])),{n,3}]//N

Table[(E^(-(2^n) Derivative[1,0][PolyLog][0,(1/2)]+(1/2) Derivative[0,1,0][LerchPhi][(1/2),0,1+n])),{n,10}]//FullSimplify

FunctionExpand[%]//FullSimplify

l=Table[(E^(-(2^n) Derivative[1,0][PolyLog][0,(1/2)]+(1/2) Derivative[0,1,0][LerchPhi][(1/2),0,1+n])),{n,10}]//FullSimplify

v7

RSolve[{g[n]==n g[n-1]^2,g[0]==1},g[n],n]

l=Table[(E^(-(2^n) Derivative[1,0][PolyLog][0,(1/2)]+(1/2) Derivative[0,1,0][LerchPhi][(1/2),0,1+n])),{n,10}]//FullSimplify

<<Utilities`Simplify`

FunctionExpand[l]//FullSimplify

(((x+1))^2)==(((A(x))^2)/(A((x/(x+1)))))

Series[Normal[(((A[x])^2)/(A[(x/(1+x))]))/.A[x_]:>Polynomial[Reverse[Array[a,5,0]],x]],{x,0,4}]//Simplify

CoefficientList[%,x]

Solve[Thread[%=={1,2,1,0,0}],Array[a,0,4]]

NestList[-#+2(Sqrt[#]/.x->x/(1+x))(1+x)&,1,2]

With[{n=10},CoefficientList[Series[Nest[-#+2(Sqrt[#]/.x->x/(1+x))(1+x)&,1,n],{x,0,n}],x]]

With[{a=a=Polynomial[Reverse[{1,2,-1,4,-21,138,-1091,10088,-106918,1279220,-17070418}],x]},
Series[((a^2)/(a/.x->x/(1+x))),{x,0,10}]
]

Exponent

N[With[{n=20},(((((g[n])/(n+2-(n^(-1))+4(n^(-2))))))^(2^(-n)))],100]//Timing

Integrals

log(σ)==(∫_0^1)(∫_0^1)-(x/((2-x y) log(x y)))dydx

Integrate[((-x)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-x)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

NIntegrate[((-x)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]

(∫_0^1)((1-X)/((X-2) log(X)))dX

Integrate[((1-X)/((-2+X) Log[X])),{X,0,1}]//Timing

NIntegrate[((1-X)/((-2+X) Log[X])),{X,0,1}]

Nested Square Roots

Sqrt[Fold[#2 Sqrt[#1]&,1,Reverse[Range[5]]]]

N[Sqrt[Fold[#2 Sqrt[#1]&,1,Reverse[Range[100]]]],20]

(Sqrt[1(Sqrt[2(Sqrt[3(Sqrt[4(Sqrt[5])])])])])

Derivatives

Derivative[1,0][PolyLog](0,(1/2))

N[Derivative[1,0][PolyLog][0,(1/2)],10]

(1/2) Derivative[0,1,0][LerchPhi]((1/2),0,1)

N[(1/2)Derivative[0,1,0][LerchPhi][(1/2),0,1],10]

expr=FullSimplify[Subtract@@{(1/2)Derivative[0,1,0][LerchPhi][(1/2),0,1],Derivative[1,0][PolyLog][0,(1/2)]}]

Developer`ZeroQ[expr]//Timing

Sums

log(σ)==(∑_(k=1)^∞)((((1/2)))^k) log(k)

NSum[((1/2))^k*Log[k],{k,Infinity}]

V6

Sum[((1/2))^k*Log[k],{k,Infinity}]

N[%,10]

V7

Sum[((1/2))^k*Log[k],{k,Infinity}]

FunctionExpand[%]

N[%,10]

log(σ)==(∑_(n=1)^∞)(((((-1))^(n+1)) (Subscript[Li, n])((1/2)))/n)

J. Borwein, pers. comm. Feb 6, 2006

V6

Sum[((-1)^(n+1)/n)*PolyLog[n, 1/2], {n, Infinity}]

NSum[((-1)^(n+1)/n)*PolyLog[n,1/2],{n,Infinity}]

V7

Sum[((-1)^(n+1)/n)*PolyLog[n, 1/2], {n, Infinity}]

log σ==(∑_(n=1)^∞)(∑_(k=1)^n)(((-1))^(k+n)) (({{n}, {k}})) log(k+1)

Block[{$MaxExtraPrecision=500},N[Sum[Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{k,n}],{n,1000}],200]
]//Timing

V6

Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{k,n}]

Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{k,n},{n,∞}]

V7

TimeConstrained[Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{k,n}],120]//Timing

TimeConstrained[Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{n,∞},{k,n}],120]//Timing

TimeConstrained[Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{k,n}],300]//Timing

$Version

N[Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{n,50},{k,n}],20]

s=Sum[(((-1))^(n+k))Binomial[n,k]Log[k+1],{n,300},{k,n}];

Block[{$MaxExtraPrecision=2000},N[s,20]]

N[-Derivative[1,0][PolyLog][0,(1/2)],20]

-Derivative[1, 0][PolyLog][0, 1/2]//N

Products

(∏_(k=1)^∞)(k^(1/(2^k)))==σ

v5.2

N[%,20]

Exp[N[Log[%%],20]]

v7

Product[(k^(1/(2^k))),{k,∞}]

(∏_k^∞)(((((k+1)/k)))^(1/(2^k)))==σ

Wallis-Type

NProduct[(((((k+1)/k)))^(1/(2^k))),{k,∞}]

V6

Product[(((((k+1)/k)))^(1/(2^k))),{k,∞}]

N[%]

...wrong; 1/correct...

V7

Product[(((((k+1)/k)))^(1/(2^k))),{k,∞}]

FullSimplify[%]

N[%]

σ==(∏_(n=1)^∞)(∏_(k=0)^n)(((k+1))^((((-1))^(k+n)) (({{n}, {k}}))))

NProduct[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{k,0,n},{n,∞}]//Timing

Product[Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{k,0,n}],{n,10}]//N

Product[Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{k,0,n}],{n,20}]//N//Timing

V7

Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{k,0,n}]

Product[Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{k,0,n}],{n,∞}]

Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{n,∞},{k,0,n}]//InputForm

N[Product[(((k+1))^((((-1))^((n+k)))Binomial[n,k])),{n,21},{k,0,n}],20]

N[E^(-Derivative[1, 0][PolyLog][0, 1/2])]