A124312 Expansion of g.f. x^3*(1 - x)/(1 - x - x^2 - x^3 - x^4 - x^5).

0, 0, 1, 0, 1, 2, 4, 8, 15, 30, 59, 116, 228, 448, 881, 1732, 3405, 6694, 13160, 25872, 50863, 99994, 196583, 386472, 759784, 1493696, 2936529, 5773064, 11349545, 22312618, 43865452, 86237208, 169537887, 333302710, 655255875, 1288199132

Offset

1,6

Comments

Second column of the n-th power of pentanacci matrix {{1,1,1,1,1},{1,0,0,0,0}, {0,1,0,0,0}, {0,0,1,0,0}, {0,0,0,1,0}} read from bottom to top gives 5 numbers starting from position n.

a(n+5) equals the number of n-length binary words avoiding runs of zeros of lengths 5i+4, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

Links

Robert Israel, Table of n, a(n) for n = 1..3407

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.

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

Maple

f:= gfun:-rectoproc({a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)-a(n+5), a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 0}, a(n), remember):
seq(f(n), n=1..30); # Robert Israel, Apr 13 2017

Mathematica

CoefficientList[Series[(x^3-x^4)/(1-x-x^2-x^3-x^4-x^5), {x, 0, 50}], x]

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0, 0] cat Coefficients(R!( x^3*(1-x)^2/(1-2*x+x^6) )); // G. C. Greubel, Aug 25 2023
(SageMath)
def A124312_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1-x)^2/(1-2*x+x^6) ).list()
A124312_list(50) # G. C. Greubel, Aug 25 2023

Crossrefs

Cf. A001591, A124311, A124313, A124314.

Sequence in context: A374682 A189101 A018089 * A068030 A251707 A251712

Adjacent sequences: A124309 A124310 A124311 * A124313 A124314 A124315

Keyword

nonn,easy

Author

Artur Jasinski, Oct 25 2006

Extensions

Edited by N. J. A. Sloane, Oct 29 2006, Jul 14 2007

Name corrected by Robert Israel, Apr 13 2017

Status

approved