A192254 0-sequence of reduction of (n^2) by x^2 -> x+1.

1, 1, 10, 26, 76, 184, 429, 941, 1994, 4094, 8208, 16128, 31169, 59393, 111818, 208330, 384620, 704408, 1280925, 2314525, 4158346, 7432606, 13223040, 23424576, 41335201, 72679969, 127373194, 222545306, 387732844, 673762744

Offset: 1

Author

Clark Kimberling, Jun 27 2011

Keywords

• nonn
• changed

Comments

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

Programs

Mathematica

c[n_] := n^2;  (* A000290 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[
  Last[Most[
    FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
   30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192254 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192255 *)
(* Peter J. C. Moses, Jun 20 2011 *)

Formula

Empirical g.f.: x*(1-3*x+9*x^2-6*x^3+2*x^4)/(1-x)/(1-x-x^2)^3. - Colin Barker, Feb 10 2012