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

%I #14 Jun 13 2015 00:53:54

%S 1,2,4,9,25,84,312,1199,4637,17906,68976,265249,1019069,3913484,

%T 15026092,57690143,221487945,850350482,3264725772,12534190569,

%U 48122302705,184755243892,709328262928,2723314511871,10455585321989,40141990468066

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

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

%C If the same reduction is applied to the sequence (x+1)^n instead of (x+2)^n, the resulting three coefficient sequences are essentially as follows:

%C A078484: constants

%C A099216: coefficients of x

%C A115390: coefficients of x^2.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,11).

%F a(n) = 7*a(n-1)-15*a(n-2)+11*a(n-3).

%F G.f.: -(5*x^2-5*x+1)/(11*x^3-15*x^2+7*x-1). [_Colin Barker_, Jul 27 2012]

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

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

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

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

%e p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4

%e p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that

%e A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).

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

%t p[n_, x_] := (x + 2)^n;

%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}] (* A192801 *)

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

%t (* A192802 *)

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

%t (* A192803 *)

%Y Cf. A192744, A192232, A192616, A192802, A192803.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jul 10 2011

%E Recurrence corrected by _Colin Barker_, Jul 27 2012