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A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

%I A168637 #26 Apr 21 2025 10:47:54
%S A168637 0,1,3,3,6,8,11,16,21,29,39,52,70,93,124,165,219,291,386,512,679,900,
%T A168637 1193,1581,2095,2776,3678,4873,6456,8553,11331,15011,19886,26344,
%U A168637 34899,46232,61245,81133,107479,142380,188614,249861,330996,438477,580859,769475
%N A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.
%D A168637 R. Pallu de la Barriere, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344
%H A168637 G. C. Greubel, <a href="/A168637/b168637.txt">Table of n, a(n) for n = 0..1000</a>
%H A168637 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1).
%F A168637 Limit_{n -> oo} a(n+1)/a(n) = A060006 (also a limiting value of A000931).
%F A168637 G.f.: x*(1 + 2*x - x^2)/((1-x)*(1 - x^2 - x^3)). [Dec 03 2009]
%F A168637 a(n) = 3*A000931(n+4) + 2*A000931(n+3) - 2. [Dec 03 2009]
%F A168637 a(n) = a(n-2) + a(n-3) + 2. - _Greg Dresden_, May 18 2020
%t A168637 LinearRecurrence[{1,1,0,-1},{0,1,3,3},50] (* or *) CoefficientList[ Series[ x*(-1-2x+x^2)/((1-x)(x^3+x^2-1)),{x,0,50}],x] (* _Harvey P. Dale_, Jun 22 2011 *)
%o A168637 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^n*[0;1;3;3])[1,1] \\ _Charles R Greathouse IV_, Jul 29 2016
%o A168637 (Magma)
%o A168637 R<x>:=PowerSeriesRing(Integers(), 60);
%o A168637 [0] cat Coefficients(R!( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) )); // _G. C. Greubel_, Apr 20 2025
%o A168637 (SageMath)
%o A168637 def A168637_list(prec):
%o A168637     P.<x> = PowerSeriesRing(ZZ, prec)
%o A168637     return P( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) ).list()
%o A168637 print(A168637_list(60)) # _G. C. Greubel_, Apr 20 2025
%Y A168637 Cf. A007307 (for a different starting vector of the Mma program).
%Y A168637 Cf. A000931, A060006.
%K A168637 nonn,easy
%O A168637 0,3
%A A168637 _Roger L. Bagula_ and _Gary W. Adamson_, Dec 01 2009
%E A168637 Precise definition and more formulas supplied by the Assoc. Editors of the OEIS, Dec 03 2009