A023424 Expansion of (1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5).

1, 3, 7, 15, 31, 57, 113, 223, 439, 863, 1695, 3333, 6553, 12883, 25327, 49791, 97887, 192441, 378329, 743775, 1462223, 2874655, 5651423, 11110405, 21842481, 42941187, 84420151, 165965647, 326279871, 641449337, 1261056193, 2479171199, 4873922247, 9581878847

Offset

0,2

Comments

Traces of successive powers of pentanacci matrix. - Artur Jasinski, Jan 05 2007

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..199 from T. D. Noe)

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

S. Saito, T. Tanaka, N. Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values , J. Int. Seq. 14 (2011) # 11.2.4, Table 3.

Eric Weisstein's World of Mathematics, Lucas n-Step Number

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Formula

a(n) = n * Sum_{k=1..n} (1/k)*Sum_{r=0..k} binomial(k,r)*Sum_{m=0..r} binomial(r,m) * Sum_{j=0..m} binomial(m,j)*binomial(j,n-m-k-j-r), n>0. - Vladimir Kruchinin, Feb 22 2011

Mathematica

LinearRecurrence[{1, 1, 1, 1, 1}, {1, 3, 7, 15, 31}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
CoefficientList[Series[(1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5), {x, 0, 50}], x] (* G. C. Greubel, Jan 01 2018 *)

Prog

(Maxima)
a(n):=n*sum(1/k*sum(binomial(k, r)*sum(binomial(r, m)*sum(binomial(m, j)*binomial(j, n-m-k-j-r), j, 0, m), m, 0, r), r, 0, k), k, 1, n);
(PARI) Vec((1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5)+O(x^100)) \\ Charles R Greathouse IV, Feb 24, 2011
(Magma) I:=[1, 3, 7, 15, 31]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 01 2018

Crossrefs

Essentially the same as A074048.

Sequence in context: A304078 A151338 A229006 * A276647 A006778 A007574

Adjacent sequences: A023421 A023422 A023423 * A023425 A023426 A023427

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved