OFFSET
1,1
COMMENTS
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
FORMULA
Empirical G.f.: -x*(3*x-1)*(8*x^2-3)/((x-1)*(x+1)*(2*x-1)*(4*x-1)). [Colin Barker, Nov 12 2012]
EXAMPLE
MATHEMATICA
Remove["Global`*"];
q[x_] := x + 2; p[n_, x_] := 1 + x^n + 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}]
(* A192465 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192466 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A192467 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 01 2011
STATUS
approved