A168639 Expansion of x*(1 + x^2 - x^3) / ( (1-x)*(1-x-x^4) ).

0, 1, 2, 4, 5, 7, 10, 15, 21, 29, 40, 56, 78, 108, 149, 206, 285, 394, 544, 751, 1037, 1432, 1977, 2729, 3767, 5200, 7178, 9908, 13676, 18877, 26056, 35965, 49642, 68520, 94577, 130543, 180186, 248707, 343285, 473829, 654016, 902724, 1246010, 1719840, 2373857, 3276582

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

• nonn
• easy

References

• R. Pallu de la Barrière, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-1).

Crossrefs

Cf. A003269, A086106, A098578.

Programs

Magma

R:=PowerSeriesRing(Integers(), 60);
[0] cat Coefficients(R!( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) )); // G. C. Greubel, Apr 20 2025

Mathematica

LinearRecurrence[{2,-1,0,1,-1}, {0,1,2,4,5}, 60] (* G. C. Greubel, Jul 28 2016 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,-1,2]^n*[0;1;2;4;5])[1,1] \\ Charles R Greathouse IV, Jul 29 2016

SageMath

def A168639_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) ).list()
print(A168639_list(60)) # G. C. Greubel, Apr 20 2025

Formula

Lim_{n -> oo} a(n+1)/a(n) = 1.38027756909761411567330169182..., see A086106.

a(n) = 2*a(n-1) -a(n-2) +a(n-4) -a(n-5). - R. J. Mathar, Dec 02 2009

a(n) = A098578(n) - A098578(n-3) + A098578(n-2). - R. J. Mathar, May 23 2013

a(n) = A003269(n+4) + A003269(n+2) - A003269(n+1) - 1. - G. C. Greubel, Apr 20 2025