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

%I A168639 #23 Apr 22 2025 06:45:40
%S A168639 0,1,2,4,5,7,10,15,21,29,40,56,78,108,149,206,285,394,544,751,1037,
%T A168639 1432,1977,2729,3767,5200,7178,9908,13676,18877,26056,35965,49642,
%U A168639 68520,94577,130543,180186,248707,343285,473829,654016,902724,1246010,1719840,2373857,3276582
%N A168639 Expansion of x*(1 + x^2 - x^3) / ( (1-x)*(1-x-x^4) ).
%D A168639 R. Pallu de la Barrière, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344.
%H A168639 G. C. Greubel, <a href="/A168639/b168639.txt">Table of n, a(n) for n = 0..1000</a>
%H A168639 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-1).
%F A168639 Lim_{n -> oo} a(n+1)/a(n) = 1.38027756909761411567330169182..., see A086106.
%F A168639 a(n) = 2*a(n-1) -a(n-2) +a(n-4) -a(n-5). - _R. J. Mathar_, Dec 02 2009
%F A168639 a(n) = A098578(n) - A098578(n-3) + A098578(n-2). - _R. J. Mathar_, May 23 2013
%F A168639 a(n) = A003269(n+4) + A003269(n+2) - A003269(n+1) - 1. - _G. C. Greubel_, Apr 20 2025
%t A168639 LinearRecurrence[{2,-1,0,1,-1}, {0,1,2,4,5}, 60] (* _G. C. Greubel_, Jul 28 2016 *)
%o A168639 (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
%o A168639 (Magma)
%o A168639 R<x>:=PowerSeriesRing(Integers(), 60);
%o A168639 [0] cat Coefficients(R!( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) )); // _G. C. Greubel_, Apr 20 2025
%o A168639 (SageMath)
%o A168639 def A168639_list(prec):
%o A168639     P.<x> = PowerSeriesRing(ZZ, prec)
%o A168639     return P( x*(1+x^2-x^3)/((1-x)*(1-x-x^4)) ).list()
%o A168639 print(A168639_list(60)) # _G. C. Greubel_, Apr 20 2025
%Y A168639 Cf. A003269, A086106, A098578.
%K A168639 nonn,easy
%O A168639 0,3
%A A168639 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009