Fibonacci Number

Author

Eric W. Weisstein
January 21, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/FibonacciNumber.nb.

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Enumeration

Sequence

Fibonacci[Range[10]]

The Da Vinci Code Sequence

Sort[{13,3,2,21,1,1,8,5}]==Fibonacci[Range[8]]

Squarefree

sqfree={6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 75, 78, 84, 90,
91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 132, 138, 144, 150,
153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 198, 200, 204, 210,
216, 220, 222, 224, 225, 228, 234, 240, 246, 250, 252, 258, 264, 270,
273, 275, 276, 280, 282, 288, 294, 300, 306, 312, 318, 324, 325, 330,
336, 342, 348, 350, 354, 360, 364, 366, 372, 375, 378, 384};

Select[sqfree,Mod[#,6]!=0&]

Select[Table[Mod[Fibonacci[6n],4],{n,300}],#!=0&]

Select[Table[Mod[Fibonacci[5n],5],{n,300}],#!=0&]

Squareful={6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 
75, 78, 84, 90, 91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 
132, 138, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 
198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240, 246, 250, 
252, 258, 264, 270, 273, 275, 276, 280, 282, 288, 294, 300, 306, 312};

Union[PrimeQ[Squareful]]

Prime

Position[Fibonacci[Range[10^3]],_?PrimeQ]//Flatten//Timing

Position[Fibonacci[Range[10^4]],_?PrimeQ]//Flatten//Timing

Plots

Real Line

FunctionExpand[Fibonacci[n]]

Plot[Fibonacci[n],{n,-8,8},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,Fibonacci[x]}]

<<Utilities`Plot`

PlotRoots[Fibonacci[n],{n,-8,5},WorkingPrecision->20]//Chop

x/.Table[FindRoot[(GoldenRatio^(2x))-Cos[π x],{x,-n+.5},WorkingPrecision->20],{n,0,8}]//Reverse//Chop

Reduce[(GoldenRatio^(2x))-Cos[π x]==0,x,Reals]

Solve[(GoldenRatio^(2x))==Cos[π x],x]

FullSimplify[Table[(((((1+Sqrt[5]))^n)-(((1-Sqrt[5]))^n))/((2^n)Sqrt[5])),{n,-8,8,1}]]

Binary Plot

<<Utilities`Plot`

Show[PixelPerfect[BasePlot[Fibonacci/@Range[255]],BlockSize->2]]

Show[PixelPerfect[BasePlot[Fibonacci/@Range[511]],BlockSize->1]]

Table[Take[IntegerDigits[Fibonacci[(2^(n+1))+(2^n)], 2],-(n+3)],{n,20}]

Digits

Digits/@(Fibonacci[10^#]&/@Range[10])//Timing

Digits/@(Fibonacci[10^#]&/@Range[9])//Timing

FullSimplify[1+Floor[Log[10,Round[(1/(Sqrt[5]))(((((1+(Sqrt[5]))/2)))^(10^n))]]],n∈Integers&&n>0]

FullSimplify[1+Log[10,(1/(Sqrt[5]))(((((1+(Sqrt[5]))/2)))^(10^n))],n∈Integers&&n>0]

Table[(((2^(1+n)) (5^n) ArcCsch[2]+Log[20])/(Log[100])),{n,5}]//N

Table[Ceiling[(2 10^n ArcCsch[2]-Log[5])/Log[100]],{n,9}]

FullSimplify[(((2 10^n ArcCsch[2]-Log[5])/Log[100])/(10^n))]

Limit[(((2 10^n ArcCsch[2]-Log[5])/Log[100])/(10^n)),n->∞]//FullSimplify

FullSimplify[FunctionExpand[Fibonacci[n]],n∈Integers&&n>0]

Limit[(1+Log[10,((-(((-(2/(1+(Sqrt[5])))))^(10^n))+((((1/2) (1+(Sqrt[5]))))^(10^n)))/(Sqrt[5]))])/10^n,n->∞]

FullSimplify[%12]

N[%,100]

Fibonacci[Range[20]]

Floor[Table[n/.FindRoot[Fibonacci[n]==10^k,{n,1},WorkingPrecision->100],{k,8}]]

Floor[Table[n/.FindRoot[Fibonacci[n]==10^k,{n,1},WorkingPrecision->500],{k,9}]]

Solve[GoldenRatio^n/Sqrt[5]==10^k,n]

Plus or minus 1

Table[Log[GoldenRatio,(2^k) (5^(k+1/2))],{k,20}]//N

Table[Log[GoldenRatio,(2^k) (5^(k+1/2))],{k,20}]//Round

Congruences

(fm=Table[Mod[Table[Fibonacci[n],{n,15}],m],{m,2,8}])//ColumnForm

eis={011655, 82115,079343,82116,82117,82116,079344};

p=Period/@Table[Mod[Table[Fibonacci[n],{n,200}],k],{k,2,8}]

MapIndexed[Join[#2+1,#1]&,Transpose[{p,eis,fm}]]

f=Fibonacci[Range[10^5]];//Timing

Table[Period[Mod[f,10^n]],{n,3}]//Timing

Table[Period[Mod[f,10^n]],{n,4}]//Timing

Convergents

ContinuedFraction[GoldenRatio,10]

Table[FromContinuedFraction[ContinuedFraction[1/GoldenRatio^2,n]],{n,10}]

ContinuedFractionK[1,{n,∞}]

ContinuedFractionK[Piecewise[{{0,n==0},{1,True}}],{n,0,∞}]

ContinuedFractionK[Piecewise[{{0,n==0},{1,True}}],{n,0,20}]//N

Determinant

FibDet[n_,k_]:=Table[Fibonacci[n+1+i k+j ],{i,0,k-1},{j,0,k-1}]

FibDet[n,1]//TraditionalForm

FibDet[n,2]//TraditionalForm

FibDet[n,3]//TraditionalForm

l=Select[Table[{n,k,Det[fibDet[n,k]]}/.Thread[{n,k}->Table[Random[Integer,{1,10}],{2}]],{100}]/.fibDet->FibDet,Last[#]≠0&]

{#,Fibonacci[First[#]+1]}&/@Select[l,#[[2]]==1&]

{#,(-1)^First[#]}&/@Select[l,#[[2]]==2&]

Sergei Markelov

FibMatrix[n_]:=Table[Which[
i==j,1,
Abs[i-j]==1,I,
True,0],{i,n},{j,n}]

Table[FibMatrix[n]//MatrixForm,{n,5}]

Det/@Table[FibMatrix[n],{n,6}]

Table[Fibonacci[n+1],{n,6}]

Limit

V5

Limit[Fibonacci[n]/Fibonacci[n-1],n->Infinity]

l=FullSimplify[FunctionExpand[Fibonacci[n]/Fibonacci[n-1]]] /. Cos[n Pi]:>(-1)^n

Limit[l,n->Infinity]

<<NumericalMath`

NLimit[l,n->Infinity]

N[l /. n->20]

ListPlot[Table[l,{n,1,20}],PlotJoined->True]

V7

Limit[Fibonacci[n]/Fibonacci[n-1],n->Infinity]

Special Function Indetities

(i^(n-1)) (Subscript[U, n-1])(-(i/2))==(Subscript[F, n])

FullSimplify[I^(n-1)ChebyshevU[n-1,-I/2],n∈Integers]

Table[{Fibonacci[n],I^(n-1)ChebyshevU[n-1,-I/2]},{n,-5,5}]

exp((∑_k^∞)(((Subscript[L, k]) (x^k))/k))==(∑_k^∞)(x^(k-1)) (Subscript[F, k])

V7

Sum[LucasL[k]((x^k)/k),{k,∞}]

FullSimplify[%]

FullSimplify[Exp[%]]

Sum[Fibonacci[k](x^(k-1)),{k,∞}]

FullSimplify[%]

V8

Sum[Fibonacci[k](x^(k-1)),{k,∞}]

Polynomial Identities

FullSimplify[Subtract@@{Fibonacci[2n],#}&/@
{
Fibonacci[n](Fibonacci[n+1]+Fibonacci[n-1]),
Fibonacci[n](Fibonacci[n]+2Fibonacci[n-1]),
Fibonacci[n](2Fibonacci[n+1]-Fibonacci[n])
},n∈Integers]

Johnson (25 Mar 2003)

With[{d=a+b-c},
FullSimplify[Subtract@@{Fibonacci[a]Fibonacci[b]-Fibonacci[c]Fibonacci[d],(-1)^r(Fibonacci[a-r]Fibonacci[b-r]-Fibonacci[c-r]Fibonacci[d-r])},(a|b|c|d|r)∈Integers]
]

Unnamed Identities

FullSimplify[Fibonacci[2n]-(Fibonacci[n+1]^2-Fibonacci[n-1]^2),n∈Integers]//Timing

FullSimplify[Fibonacci[3n]-(Fibonacci[n+1]^3+Fibonacci[n]^3-Fibonacci[n-1]^3),n∈Integers]//Timing

FullSimplify[Fibonacci[n+1]^2-(4Fibonacci[n]Fibonacci[n-1]+Fibonacci[n-2]^2),n∈Integers]//Timing

Brousseau's Identity

FullSimplify[Fibonacci[n+1]^2-4Fibonacci[n]Fibonacci[n-1]-Fibonacci[n-2]^2]

Catalan's Identity

FullSimplify[Fibonacci[n]^2-Fibonacci[n+r]Fibonacci[n-r]-(-1)^(n-r)Fibonacci[r]^2]

FunctionExpand[%]

FullSimplify[%14,(r|n)∈Integers]

d'Ocagne's Identity

FullSimplify[Fibonacci[m]Fibonacci[n+1]-Fibonacci[n]Fibonacci[m+1]-(-1)^(n)Fibonacci[m-n]]

FunctionExpand[%12]

FullSimplify[%15,(m|n)∈Integers]

Gelin-Cesàro Identity

FunctionExpand[Fibonacci[n]^4-Fibonacci[n-2]Fibonacci[n-1]Fibonacci[n+1]Fibonacci[n+2]]

FullSimplify[%,n∈Integers]

Pythagorean Triple

4 (Subscript[F, k+1]^2) (Subscript[F, k+2]^2)==((((Subscript[F, k+1]^2)+(Subscript[F, k+2]^2)))^2)+(Subscript[F, k]^2) (Subscript[F, k+3]^2)

Subtract@@{Plus@@({Fibonacci[k]Fibonacci[k+3],2Fibonacci[k+1]Fibonacci[k+2]}^2),(Fibonacci[k+1]^2+Fibonacci[k+2]^2)^2}

FullSimplify[%]

Product

f[n_]:=Product[Fibonacci[k],{k,n}]+1

V5

t=Table[f[n],{n,50}];

A052359

A053408

Position[t,_?PrimeQ]//Flatten

A053413

Select[t,PrimeQ]

V7

f[n]

Generating Functions

Generating Function

Plot[-(x/(-1+x+(x^2))),{x,-GoldenRatio,GoldenRatio-1},PlotStyle->Red]

V5

GeneratingFunction[{a[n]==a[n-1]+a[n-2],a[1]==a[2]==1},a[n],n,z]//Simplify

FullSimplify[Sum[Fibonacci[n]x^n,{n,0,∞}]]

FullSimplify[Sum[Fibonacci[n]x^n,{n,∞}]]

V7

RSolve[{a[n]==a[n-1]+a[n-2],a[1]==a[2]==1},a[n],n]

GeneratingFunction[Fibonacci[n],n,z]

FullSimplify[Sum[Fibonacci[n]x^n,{n,0,∞}]]

FullSimplify[Sum[Fibonacci[n]x^n,{n,∞}]]

SeriesCoefficient[x/(1-x-x^2),{x,0,n}]//InputForm

FullSimplify[%,n∈Integers]//InputForm

V8

Sum[Fibonacci[n]x^n,{n,∞}]//FullSimplify

GeneratingFunction[Fibonacci[n],n,z]

ExponentialGeneratingFunction

Sum[Fibonacci[n]((x^n)/(n!)),{n,∞}]//FullSimplify

(2/(Sqrt[5]))Exp[x/2]Sinh[Sqrt[5]x/2]-(((E^(-((2 x)/(1+(Sqrt[5]))))) (-1+(E^((Sqrt[5]) x))))/(Sqrt[5]))//FullSimplify

ExponentialGeneratingFunction[Fibonacci[n],n,z]//FullSimplify

DirichletGeneratingFunction

Sum[((Fibonacci[n])/(n^s)),{n,∞}]

(1/(Sqrt[5]))Sum[((((GoldenRatio^n)-(((-GoldenRatio))^(-n))))/(n^s)),{n,∞}]

Sum[((Fibonacci[n])/(n^s)),{n,∞},Assumptions->s>0]

Sum[((Fibonacci[n])/(n^s)),{n,∞},Assumptions->s>1]

With[{s=10},Sum[((Fibonacci[n])/(n^s)),{n,∞}]]

Sum[((Fibonacci[n])/(n^s)),{n,20}]

Plot[Sum[((Fibonacci[n])/(n^s)),{n,50}],{s,0,10}]

Lambert Series

Sum[((Fibonacci[n](x^n))/(1-(x^n))),{n,∞}]

(1/(Sqrt[5]))Sum[((GoldenRatio^n)-(((-GoldenRatio))^(-n)))((x^n)/(1-(x^n))),{n,∞}]

(1/(Sqrt[5]))Sum[(GoldenRatio^n)((x^n)/(1-(x^n))),{n,∞}]

(1/(Sqrt[5]))Sum[(((-GoldenRatio))^(-n))((x^n)/(1-(x^n))),{n,∞}]

Factor[((ϕ^n)-(((-ϕ))^(-n)))((x^n)/(1-(x^n)))]

Finite Sums

(∑_(k=0)^n)(Subscript[F, k]) (Binomial[n, k])==(Subscript[F, 2 n])

V7

Sum[Binomial[n,k]Fibonacci[k],{k,0,n}]

FullSimplify[%,n∈Integers&&n>0]

Table[{((-((((1/2) (3-(Sqrt[5]))))^n)+((((1/2) (3+(Sqrt[5]))))^n))/(Sqrt[5])),Fibonacci[2n]},{n,0,5}]//FullSimplify

V8

Sum[Binomial[n,k]Fibonacci[k],{k,0,n}]

(∑_(k=0)^n)(Subscript[(2^k)F, k]) (Binomial[n, k])==(Subscript[F, 3 n])

V7

Sum[2^k Binomial[n,k]Fibonacci[k],{k,0,n}]

FullSimplify[%,n∈Integers&&n>0]

Table[{((-(((2-(Sqrt[5])))^n)+(((2+(Sqrt[5])))^n))/(Sqrt[5])),Fibonacci[3n]},{n,0,5}]//FullSimplify

V8

Sum[2^k Binomial[n,k]Fibonacci[k],{k,0,n}]

(∑_(k=0)^(n/2))(Binomial[n-k, k])==(Subscript[F, n+1])

V5

s=Sum[Binomial[n-k,k],{k,0,n/2}]

Table[s,{n,5}]

Table[Fibonacci[n+1],{n,5}]

V7

Sum[Binomial[n-k,k],{k,0,n/2}]

FullSimplify[%]

FullSimplify[(1/2) (Fibonacci[n]+LucasL[n])-Fibonacci[n+1]]

(∑_(k=0)^n)(Binomial[n-k, k])==(Subscript[F, n+1])

Shallow Diagonal

Table[Sum[Binomial[n-k,k],{k,0,n}],{n,10}]

V5

s=Sum[Binomial[n-k,k],{k,0,n}]

Wrong

N[Table[s,{n,5}],20]

FullSimplify[s,{n>0,n∈Integers}]

Table[(I^n) ChebyshevU[n,-(I/2)],{n,10}]

FunctionExpand[s]

FullSimplify[%]

FullSimplify[ComplexExpand[%]] /. Exp[I n Pi]:>(-1)^n

FullSimplify[%,{n>0,n∈Integers}]

V7

Sum[Binomial[n-k,k],{k,0,n}]

(∑_k^n)(Fibonacci[k])==(Subscript[(Fibonacci[n]), +2])-1

V5

FullSimplify[Sum[Fibonacci[n],{n,k}],k∈Integers&&k>0]

ListPlot[Table[-1+(((2^(-k)) ((((1-(Sqrt[5])))^(1+k))+2 (((1+(Sqrt[5])))^k) (2+(Sqrt[5]))))/(5+(Sqrt[5]))),{k,10}]]

Limit[-1+(((2^(-k)) ((((1-(Sqrt[5])))^(1+k))+2 (((1+(Sqrt[5])))^k) (2+(Sqrt[5]))))/(5+(Sqrt[5]))),k->∞]

Series[x/(1-x-x^2),{x,0,10}]

<<DiscreteMath`RSolve`

SeriesTerm[x/(1-x-x^2),{x,0,n}]

FullSimplify[FunctionExpand[Fibonacci[n]],n∈Integers]

FullSimplify[Subtract@@{Sum[Fibonacci[k],{k,n}],Fibonacci[n+2]-1},n∈Integers]

V6

FullSimplify[Sum[Fibonacci[n],{n,k}],k∈Integers&&k>0]

V7

Sum[Fibonacci[k],{k,n}]

(∑_(k=0)^n)(Subscript[F, 2 k+1])==(Subscript[F, 2 n+2])

V5

FullSimplify[Subtract@@{Sum[Fibonacci[2k+1],{k,0,n}],Fibonacci[2n+2]},n∈Integers]

V7

Sum[Fibonacci[2k+1],{k,0,n}]

(∑_(k=0)^n)(Subscript[F, 2 k])==(Subscript[F, 2 n+1])-1

V5

FullSimplify[Subtract@@{Sum[Fibonacci[2k],{k,0,n}],Fibonacci[2n+1]-1},n∈Integers]

V7

Sum[Fibonacci[2k],{k,0,n}]

(∑_(k=1)^n)(Subscript[F, k]^2)=(Subscript[F, n]) (Subscript[F, n+1])

v4.1

Sum[Fibonacci[k]^2,{k,n}]-Fibonacci[n]Fibonacci[n+1]

FullSimplify[Table[-1+(((((2 (1+(Sqrt[5]))))^(-2 n)) ((2^(1+4 n))+(((1+(Sqrt[5])))^(4 n)) (3+(Sqrt[5]))))/(5+(Sqrt[5])))-Fibonacci[n] Fibonacci[1+n],{n,0,6}]]

...wrong...

v4.2

s=Sum[Fibonacci[k]^2,{k,n}]

...wrong...

FullSimplify[Table[Fibonacci[n]Fibonacci[n+1]-s,{n,5}]]

v5.0

s=Sum[Fibonacci[k]^2,{k,n}]

FullSimplify[Table[Fibonacci[n]Fibonacci[n+1]-s,{n,5}]]

FullSimplify[Sum[Fibonacci[k]^2,{k,n}]-Fibonacci[n]Fibonacci[n+1],n∈Integers]//Timing

....correct...

v5.2

FullSimplify[Subtract@@{Sum[Fibonacci[k]^2,{k,n}],Fibonacci[n]Fibonacci[n+1]},n∈Integers]

V7

Sum[Fibonacci[k]^2,{k,n}]

(∑_(i=0)^(⌊(k-1)/2⌋))(5^i) (Binomial[k, 2 i+1]) (Subscript[F, n]^(2 i+1)) (Subscript[L, n]^(-2 i+k-1))==(Subscript[F, k n])

V5

FullSimplify[Floor[(k-1)/2],k∈Integers&&Mod[k,2]==#]&/@{0,1}

Do *not* do this; the upper index *must* be an integer

Table[{
{k,n},
N[{Sum[Binomial[k,2i+1]5^i Fibonacci[n]^(2i+1)Lucas[n]^(k-1-2i),{i,0,Floor[(k-1)/2]}]/2^(k-1),Fibonacci[k n]},20]}/.{k->Random[Integer,{0,10}],n->Random[Integer,{0,10}]},{5}]//ColumnForm

k even

k'=(k-2)/2

id1=Sum[Binomial[(2kp+2)-i-1,i](((-1))^(i(n+1)))((Lucas[n])^((2kp+2)-1-2i)),{i,0,kp}]Fibonacci[n]/.kp->((k-2)/2)

FullSimplify[id1,(k|n)∈Integers&&n>0&&k>0]

FullSimplify[Table[{{k,n},{id,Fibonacci[k n]}}/.{k->Random[Integer,10],n->Random[Integer,10]},{10}]]//ColumnForm

k odd

k'=(k-2)/2

id2=Sum[Binomial[k-i-1,i](((-1))^(i(n+1)))((Lucas[n])^(k-1-2i))/.k->2kp+1,{i,0,kp}]Fibonacci[n]/.kp->((k-1)/2)

FullSimplify[id2,(k|n)∈Integers&&n>0&&k>0]

FullSimplify[Table[{{k,n},{id2,Fibonacci[k n]}}/.{k->Random[Integer,10],n->Random[Integer,10]},{10}]]//ColumnForm

V7

Sum[Binomial[k,2i+1]5^i Fibonacci[n]^(2i+1)LucasL[n]^(k-1-2i),{i,0,Floor[(k-1)/2]}]/2^(k-1)

FullSimplify[%,(n|k)∈Integers&&n>0&&k>0]

Table[Fibonacci[k n]-%,{k,0,5},{n,0,5}]//FullSimplify

(∑_k^n)(Subscript[F, k]) (Subscript[F, n-k])==(1/5) (n (Subscript[L, n])-(Subscript[F, n]))

V7

Sum[Fibonacci[k]Fibonacci[n-k],{k,n}]

FullSimplify[%,n∈Integers&&n>0]

Table[{(1/(5 (5+(Sqrt[5]))))(((-2 (1+(Sqrt[5]))))^(-n)) ((1+(Sqrt[5])) ((4^n)-(((-2 (3+(Sqrt[5]))))^n))+(5+(Sqrt[5])) ((4^n)+(((-2 (3+(Sqrt[5]))))^n)) n),(1/5)(n LucasL[n]-Fibonacci[n])},{n,0,10}]//FullSimplify

(Subscript[F, n k])=(∑_(i=0)^k)(({{k}, {i}})) (Subscript[F, i]) (Subscript[F, n]^i) (Subscript[F, n-1]^(k-i))

Wrong in v4.2

s=Sum[Binomial[k,i]Fibonacci[i]Fibonacci[n]^i Fibonacci[n-1]^(k-i),{i,0,k}]

FullSimplify[Table[s-Fibonacci[k n],{k,7},{n,5}]]

Correct in v5

Careful for n=1, at which point the sum contains the undefined ((Subscript[F, 0])^0)

s=Sum[Binomial[k,i]Fibonacci[i]Fibonacci[n]^i Fibonacci[n-1]^(k-i),{i,0,k}]

FullSimplify[Table[s-Fibonacci[k n],{k,7},{n,5}]]

s2=Sum[Binomial[k,i]Fibonacci[-i]Fibonacci[n]^i Fibonacci[n+1]^(k-i),{i,0,k}]

FullSimplify[Table[s2-Fibonacci[k n],{k,7},{n,5}]]

s3=Sum[Binomial[k,i]Fibonacci[i](-1)^(i+1)Fibonacci[n]^i Fibonacci[n+1]^(k-i),{i,0,k}]

s2==s3

V7

Sum[Binomial[k,i]Fibonacci[i]Fibonacci[n]^i Fibonacci[n-1]^(k-i),{i,0,k}]

FullSimplify[%,(n|k)∈Integers&&n>1&&k>0]

Table[Fibonacci[k n]-%,{n,2,5},{k,0,5}]//FullSimplify

(Subscript[F, k n+c])=(∑_(i=0)^k)(({{k}, {i}})) (Subscript[F, c-i]) (Subscript[F, n]^i) (Subscript[F, n+1]^(k-1))

Correct in v5

s=Sum[Binomial[k,i]Fibonacci[c-i]Fibonacci[n]^i Fibonacci[n+1]^(k-i),{i,0,k}]

FullSimplify[s,(k|n|c)∈Integers]

FullSimplify[Table[s-Fibonacci[k n+c],{k,4},{n,4},{c,3}]]

Infinite Sums

(∑_k^∞)(1/(Subscript[F, k]))==3.35988...

TimeConstrained[Sum[(1/(Fibonacci[k])),{k,∞}],60]//Timing

NSum[(1/(Fibonacci[k])),{k,∞}]

V8

TimeConstrained[Sum[(1/(Fibonacci[k])),{k,∞}],60]//Timing

(∑_(n=1)^∞)(Fibonacci[n])==∞

V5

Sum[Fibonacci[n],{n,∞}]

V7

Sum[Fibonacci[n],{n,∞}]

(∑_(n=1)^∞)((((-1))^(n+1))/((Subscript[F, n]) (Subscript[F, n+1])))+1==ϕ

V5

1+Sum[(-1)^(n+1)/Fibonacci[n]/Fibonacci[n+1],{n,∞}]

N[%,20]

V7

1+Sum[(-1)^(n+1)/Fibonacci[n]/Fibonacci[n+1],{n,∞}]

(Subscript[F, n])==(1/(2^(n-1)))(∑_(k=0)^∞)(5^k) (Binomial[n, 2 k+1])

Table[Sum[5^k Binomial[n,2k+1],{k,0,∞}]/2^(n-1),{n,10}]

v4.2

FullSimplify[Sum[5^k Binomial[n,2k+1],{k,0,n/2}]/2^(n-1)]

FullSimplify[Sum[5^k Binomial[n,2k+1],{k,0,n/2}]/2^(n-1)-Fibonacci[n],n∈Integers]

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

v5.0

FullSimplify[Sum[5^k Binomial[n,2k+1],{k,0,n/2}]/2^(n-1)]

FullSimplify[Sum[5^k Binomial[n,2k+1],{k,0,n/2}]/2^(n-1)-Fibonacci[n],n∈Integers]

Table[Sum[5^k Binomial[n,2k+1],{k,0,∞}]/2^(n-1),{n,10}]

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

V7

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

(∑_(n=0)^∞)((Subscript[F, n])/(10^n))==(10/89)

Sum[Fibonacci[n]/10^n,{n,0,∞}]

(∑_(n=0)^∞)((Subscript[F, n])/(10^(n+1)))==(1/89)

Sum[((Fibonacci[n])/(10^(n+1))),{n,0,∞}]

(∑_(n=1)^∞)((((-1))^n)/((Fibonacci[n]) (Fibonacci[n+2])))==2-(Sqrt[5])

V5

{#,N[#]}&@{Sum[(-1)^n/Fibonacci[n]/Fibonacci[n+2],{n,∞}],2-Sqrt[5]}

{#,N[#]}&@{Sum[(-1)^(n+1)/Fibonacci[n+1]/Fibonacci[n+2],{n,Infinity}],1/GoldenRatio^2}

{#,N[#]}&@{Sum[1/Fibonacci[2n]/Fibonacci[2n+2],{n,Infinity}],1/GoldenRatio^2}

{#,N[#]}&@{Sum[Fibonacci[n]/Fibonacci[n+1]/Fibonacci[n+2],{n,∞}],1}

{#,N[#]}&@{Sum[1/Fibonacci[2^n],{n,0,∞}],(7-Sqrt[5])/2}

V7

Sum[(-1)^n/Fibonacci[n]/Fibonacci[n+2],{n,∞}]

NSum[(-1)^n/Fibonacci[n]/Fibonacci[n+2],{n,∞}]

2-Sqrt[5]//N

(∑_n^∞)((((-1))^(n+1))/((Subscript[F, n+1]) (Subscript[F, n+2])))==(1/(ϕ^2))

V7

Sum[((((-1))^(n+1))/(Fibonacci[n+1]Fibonacci[n+2])),{n,∞}]

(∑_n^∞)(1/((Subscript[F, 2 n]) (Subscript[F, 2 n+2])))==(1/(ϕ^2))

V7

TimeConstrained[Sum[(1/(Fibonacci[2n]Fibonacci[2n+2])),{n,∞}],60]//Timing

NSum[(1/(Fibonacci[2n]Fibonacci[2n+2])),{n,∞}]

1/GoldenRatio^2//N

V8

TimeConstrained[Sum[(1/(Fibonacci[2n]Fibonacci[2n+2])),{n,∞}],60]//Timing

(∑_n^∞)((Subscript[F, n])/((Subscript[F, n+1]) (Subscript[F, n+2])))==1

V7

TimeConstrained[Sum[((Fibonacci[n])/(Fibonacci[n+1]Fibonacci[n+2])),{n,∞}],60]//Timing

NSum[((Fibonacci[n])/(Fibonacci[n+1]Fibonacci[n+2])),{n,∞}]

V8

TimeConstrained[Sum[((Fibonacci[n])/(Fibonacci[n+1]Fibonacci[n+2])),{n,∞}],60]//Timing

(∑_(n=0)^∞)(1/(Subscript[F, (2^n)]))==(1/2) (7-(Sqrt[5]))

V7

Sum[(1/(Fibonacci[(2^n)])),{n,0,∞}]//Timing

NSum[(1/(Fibonacci[(2^n)])),{n,0,∞}]

N[(1/2)(7-(Sqrt[5]))]

V8

Sum[(1/(Fibonacci[(2^n)])),{n,0,∞}]

(∑_(n=0)^∞)(Fibonacci[n]) (x^n)==-(x/((x^2)+x-1))

V8

Sum[Fibonacci[n] x^n, {n, 0, Infinity}]