A005923 From solution to a difference equation.

1, 3, 13, 81, 673, 6993, 87193, 1268361, 21086113, 394368993, 8195330473, 187336699641, 4671623344753, 126204511859793, 3671695236949753, 114451527759954921, 3805443567253430593, 134436722612325267393, 5028681509898733705033, 198550708258762398282201

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Binomial transform of A000557. - Vladimir Reshetnikov, Oct 29 2015

References

• Anthony G. Shannon and Richard L. Ollerton. "A note on Ledin's summation problem." The Fibonacci Quarterly 59:1 (2021), 47-56. See p. 49.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
• P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
• Prabha Sivaraman Nair, A note on alternating weighted sums of Fibonacci numbers, Math. Montisnigri (2024) Vol. LX, 32-49. See p. 38.

Crossrefs

Cf. A000556, A000557.

Programs

Mathematica

Round@Table[Sum[Binomial[n, k] (-1)^k (PolyLog[-k, 1-GoldenRatio] - PolyLog[-k, GoldenRatio])/Sqrt[5] , {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

Formula

E.g.f.: exp(x)/(1-2*sinh(x)). - Sander Zwegers (s.zwegers(AT)hetnet.nl), Jun 28 2007

E.g.f.: 1/( U(0) -1 ) where U(k) = 1 + 1/(2^k - 2*x*4^k/(2*x*2^k - (k+1)/U(k+1) )); (continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012

a(n) ~ n! * phi / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio. - Vaclav Kotesovec, Nov 27 2017

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k + 1) * binomial(n,k) * (2^k + 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020

a(n) = A000556(n) + A000557(n) for n>0. - Greg Dresden, May 13 2022

Extensions

More terms from Vladeta Jovovic, Nov 23 2001