A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.

1, 2, 4, 9, 25, 84, 312, 1199, 4637, 17906, 68976, 265249, 1019069, 3913484, 15026092, 57690143, 221487945, 850350482, 3264725772, 12534190569, 48122302705, 184755243892, 709328262928, 2723314511871, 10455585321989, 40141990468066

Offset: 0

Author

Clark Kimberling, Jul 10 2011

Keywords

• nonn
• easy

Comments

For discussions of polynomial reduction, see A192232 and A192744.

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:

A078484: constants

A099216: coefficients of x

A115390: coefficients of x^2.

Examples

			The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+2 -> x+2
p(3,x)=x^2+4x+4 -> x^2+1
p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-15,11).

Crossrefs

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

Programs

Mathematica

q = x^3; s = x^2 + x + 1; z = 40;
p[n_, x_] := (x + 2)^n;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192801 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192802 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A192803 *)

Formula

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

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

Extensions

Recurrence corrected by Colin Barker, Jul 27 2012