A135961 -id:A135961 - cp's OEIS frontend

A135741 E.g.f.: A(x) = Sum_{n>=0} exp(Fibonacci(n)x) x^n/n!.

1, 1, 3, 7, 19, 71, 347, 2115, 16395, 164071, 2099991, 34138963, 706636219, 18658538939, 627640554659, 26870678088831, 1464622985216331, 101659649883242575, 8983370274289495947, 1010460953826963080371

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 7*x^3/3! + 19*x^4/4! + 71*x^5/5! + 347*x^6/6! +...
where
A(x) = 1 + exp(x)*x + exp(x)*x^2/2! + exp(2*x)*x^3/3! + exp(3*x)*x^4/4! + exp(5*x)*x^5/5! + exp(8*x)*x^6/6! + exp(13*x)*x^7/7! + exp(21*x)*x^8/8! +...

G. C. Greubel, Table of n, a(n) for n = 0..125

Cf. A135961, A219781.

Flatten[{1,Table[Sum[Binomial[n,k]*Fibonacci[k]^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 14 2014 *)

{a(n)=sum(k=0,n,binomial(n,k)*fibonacci(k)^(n-k))}

{a(n)=n!*polcoeff(sum(k=0,n,exp(fibonacci(k)*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(sum(k=0, n, x^k/(1-fibonacci(k)*x +x*O(x^n))^(k+1)), n)}

a(n) = Sum_{k=0..n} C(n,k) * Fibonacci(k)^(n-k).
O.g.f.: Sum_{n>=0} x^n/(1 - Fibonacci(n)*x)^(n+1).
a(n) ~ c * 2^(n+1/2) * ((1+sqrt(5))/2)^(n^2/4) / (sqrt(Pi*n) * 5^(n/4)), where c = Sum_{k=-infinity..infinity} 5^(k/2)*((1+sqrt(5))/2)^(-k^2) = 3.5769727481316948565395...(see A219781) if n is even, and c = Sum_{k=-infinity..infinity} 5^((k+1/2)/2)*((1+sqrt(5))/2)^(-(k+1/2)^2) = 3.5769727390073366345992... if n is odd. - Vaclav Kotesovec, Jul 14 2014

A210342 Row sums of triangle A210341.

Original entry on oeis.org

1, 2, 4, 9, 24, 78, 317, 1636, 10752, 89871, 955536, 12930172, 222618065, 4874855542, 135781292308, 4811103270053, 216847500834512, 12432143862756778, 906625645142897789, 84102571511631809864, 9923979699312024569440, 1489546408205976627946331

Offset: 0

Emanuele Munarini, Mar 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A210341.

[&+[Fibonacci(n-k+2)^k: k in [0..n]]: n in [0..21]]; // Bruno Berselli, Mar 28 2012

Table[Sum[Fibonacci[n - k + 2]^k, {k, 0, n}], {n, 0, 100}]

makelist(sum(fib(n-k+2)^k,k,0,n),n,0,12);

G.f.: Sum_{k>=0} x^k/(1-Fibonacci(k+2)*x).
a(n) = [x^n] Sum_{k=0..n} x^k/(1-Fibonacci(k+2)*x).
a(n) = A135961(n+2)-1. - Vaclav Kotesovec, Jan 05 2013

A187780 Sum_{k=0..n} Lucas(k)^(n-k).

Original entry on oeis.org

1, 3, 6, 13, 31, 84, 271, 1111, 6096, 44965, 434321, 5388944, 85434621, 1727597731, 44466614106, 1455616862597, 60619117448531, 3211943842710212, 216483614502128251, 18558646821817827015, 2023790814160269113876, 280732940929438329958733, 49535201863823517417076181

Offset: 0

Vaclav Kotesovec, Jan 05 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000032, A135961.

Table[Sum[LucasL[k]^(n-k), {k, 0, n}], {n, 0, 20}]
(* constants: *)
ceven = N[Sum[((1+Sqrt[5])/2)^(-k^2), {k, -Infinity, +Infinity}], 50]
codd = N[Sum[((1+Sqrt[5])/2)^(-(k+1/2)^2), {k, -Infinity, +Infinity}], 50]

Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=sum(k=0, n, Lucas(k)^(n-k))
for(n=0,21,print1(a(n),", ")) \\ Paul D. Hanna, Jan 05 2013

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-k^2) = 2.555093469444518777230568... if n is even and c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-(k+1/2)^2) = 2.555093456793304790966994... if n is odd

G.f.: A(x) = Sum_{n>=0} x^n/(1 - Lucas(n)*x).

A219781 Decimal expansion of Sum_{n = -infinity..infinity} 5^(n/2)*((1+sqrt(5))/2)^(-n^2).

Original entry on oeis.org

3, 5, 7, 6, 9, 7, 2, 7, 4, 8, 1, 3, 1, 6, 9, 4, 8, 5, 6, 5, 3, 9, 5, 1, 9, 7, 8, 4, 8, 5, 6, 3, 7, 4, 5, 3, 7, 5, 2, 8, 4, 0, 5, 3, 6, 5, 2, 3, 5, 2, 2, 2, 3, 0, 7, 7, 3, 6, 9, 7, 2, 7, 3, 7, 6, 1, 2, 6, 1, 7, 5, 8, 2, 7, 8, 1, 1, 1, 2, 7, 1, 7, 5, 7, 3, 0, 2

Offset: 1

Vaclav Kotesovec, Nov 29 2012

Constant is associated with A135961.

			3.5769727481316948565395...

Cf. A135961, A135741.

RealDigits[N[Sum[5^(k/2)*((1+Sqrt[5])/2)^(-k^2), {k,-Infinity,Infinity}], 131], 10, 130][[1]]

cp's OEIS Frontend

A135741 E.g.f.: A(x) = Sum_{n>=0} exp(Fibonacci(n)*x) * x^n/n!.

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A210342 Row sums of triangle A210341.

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A187780 Sum_{k=0..n} Lucas(k)^(n-k).

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A219781 Decimal expansion of Sum_{n = -infinity..infinity} 5^(n/2)*((1+sqrt(5))/2)^(-n^2).

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A135741 E.g.f.: A(x) = Sum_{n>=0} exp(Fibonacci(n)x) x^n/n!.