A007552 Exponentiation of Fibonacci numbers. (Formerly M2860)

1, 3, 10, 42, 204, 1127, 6924, 46704, 342167, 2700295, 22799218, 204799885, 1947993126, 19540680497, 206001380039, 2275381566909, 26261810071925, 315969045744894, 3954454344433658, 51382626410402336, 691956435942841207

Offset

1,2

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002. [pLink to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Formula

E.g.f.: exp(exp(x/2)*(sqrt(5)*cosh(x*sqrt(5)/2)+sinh(x*sqrt(5)/2))/sqrt(5)-1)-1. - Vladimir Kruchinin, Feb 27 2015

Maple

f:= proc(n) option remember; `if`(n<2, 1, f(n-1) +f(n-2)) end: a:= proc(n) option remember; f(n) +add(binomial(n-1, k-1) *f(k) *a(n-k), k=1..n-1) end: seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

Mathematica

f[n_] := f[n] = If[n<2, 1, f[n-1]+f[n-2]]; a[n_] := a[n] = f[n]+Sum [Binomial[n-1, k-1]*f[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)

Prog

(PARI) Vec(serlaplace(exp( serconvol(Ser(1/(1-x-x^2)), exp(x))-1)))
/* ==> [1, 1, 3, 10, 42, 204, 1127, 6924, 46704, ...] (note offset 0) */
/* Joerg Arndt, Jun 16 2010 */

Crossrefs

Cf. A006701, A256180.

Sequence in context: A361955 A300632 A300511 * A125274 A030903 A030816

Adjacent sequences: A007549 A007550 A007551 * A007553 A007554 A007555

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved