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A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.

%I A192805 #11 Jun 13 2015 00:53:54
%S A192805 1,1,1,2,3,6,12,25,53,113,242,519,1114,2392,5137,11033,23697,50898,
%T A192805 109323,234814,504356,1083305,2326829,4997793,10734754,23057167,
%U A192805 49524466,106373552,228479649,490751217,1054084065,2264066146,4862985491
%N A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1.  See Comments.
%C A192805 For discussions of polynomial reduction, see A192232 and A192744.
%H A192805 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1,-1).
%F A192805 a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4).
%F A192805 G.f.: -(1+x)*(2*x-1) / ( (x-1)*(x^3+2*x^2+x-1) ). - _R. J. Mathar_, May 06 2014
%F A192805 a(n)-a(n-1) = A002478(n-3). - _R. J. Mathar_, May 06 2014
%e A192805 The first five polynomials p(n,x) and their reductions:
%e A192805 p(1,x)=1 -> 1
%e A192805 p(2,x)=x+1 -> x+1
%e A192805 p(3,x)=x^2+x+1 -> x^2+x+1
%e A192805 p(4,x)=x^3+x^2+x+1 -> 2x^2+3x+2
%e A192805 p(5,x)=x^4+x^3+x^2+x+1 -> 5x^2+6*x+3, so that
%e A192805 A192805=(1,1,1,2,3,...), A002478=(0,1,1,3,6,...), A077864=(0,0,1,2,5,...).
%t A192805 q = x^3; s = x^2 + 2 x + 1; z = 40;
%t A192805 p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
%t A192805 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192805 reduce[{p1_, q_, s_, x_}] :=
%t A192805 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192805        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192805 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192805 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192805   (* A192805 *)
%t A192805 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192805   (* A002478  *)
%t A192805 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192805   (* A077864 *)
%Y A192805 Cf. A192744, A192232, A002478.
%K A192805 nonn
%O A192805 0,4
%A A192805 _Clark Kimberling_, Jul 10 2011