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Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.
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%I #11 Jun 13 2015 00:53:54

%S 1,1,1,2,3,6,12,25,53,113,242,519,1114,2392,5137,11033,23697,50898,

%T 109323,234814,504356,1083305,2326829,4997793,10734754,23057167,

%U 49524466,106373552,228479649,490751217,1054084065,2264066146,4862985491

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

%C For discussions of polynomial reduction, see A192232 and A192744.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1,-1).

%F a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4).

%F G.f.: -(1+x)*(2*x-1) / ( (x-1)*(x^3+2*x^2+x-1) ). - _R. J. Mathar_, May 06 2014

%F a(n)-a(n-1) = A002478(n-3). - _R. J. Mathar_, May 06 2014

%e The first five polynomials p(n,x) and their reductions:

%e p(1,x)=1 -> 1

%e p(2,x)=x+1 -> x+1

%e p(3,x)=x^2+x+1 -> x^2+x+1

%e p(4,x)=x^3+x^2+x+1 -> 2x^2+3x+2

%e p(5,x)=x^4+x^3+x^2+x+1 -> 5x^2+6*x+3, so that

%e A192805=(1,1,1,2,3,...), A002478=(0,1,1,3,6,...), A077864=(0,0,1,2,5,...).

%t q = x^3; s = x^2 + 2 x + 1; z = 40;

%t p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A192805 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t (* A002478 *)

%t u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]

%t (* A077864 *)

%Y Cf. A192744, A192232, A002478.

%K nonn

%O 0,4

%A _Clark Kimberling_, Jul 10 2011