A005923 From solution to a difference equation. (Formerly M2953)

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

Offset

0,2

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

Table of n, a(n) for n=0..19.

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.

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

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 *)

Crossrefs

Cf. A000556, A000557.

Sequence in context: A184972 A160882 A135921 * A335588 A089461 A000684

Adjacent sequences: A005920 A005921 A005922 * A005924 A005925 A005926

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Nov 23 2001

Status

approved