A192475 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^(n+1)+x^(2n).

2, 5, 11, 26, 63, 157, 398, 1021, 2639, 6854, 17855, 46601, 121770, 318421, 833027, 2179906, 5705471, 14934533, 39094934, 102345101, 267932007, 701437390, 1836358271, 4807602001, 12586390418, 32951476517, 86267889083, 225851947946

Offset

1,1

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Links

Table of n, a(n) for n=1..28.

Formula

Empirical G.f.: -x*(x^3-3*x^2-3*x+2)/((x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Nov 12 2012

a(n) = F(2n) + F(n+1), where F(n) is A000045. - Carl Najafi, May 06 2014

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+2x^2 -> 3+2x
p(2,x)=1+x^3+x^4 -> 4+5x
p(3,x)=1+x^4+x^6 -> 8+11x
p(4,x)=1+x^5+x^8 -> 17+26x.
From these, read
A192474=(3,4,8,17,...) and A192475=(2,5,11,26,...)

Mathematica

q[x_] := x + 1;
p[n_, x_] := 1 + x^(n + 1) + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
  (* A192474 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A192475 *)

Crossrefs

Cf. A192232, A192475.

Sequence in context: A247471 A082397 A051286 * A192400 A308154 A182053

Adjacent sequences: A192472 A192473 A192474 * A192476 A192477 A192478

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved