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

%I A192254 #14 Aug 23 2025 08:03:30
%S A192254 1,1,10,26,76,184,429,941,1994,4094,8208,16128,31169,59393,111818,
%T A192254 208330,384620,704408,1280925,2314525,4158346,7432606,13223040,
%U A192254 23424576,41335201,72679969,127373194,222545306,387732844,673762744
%N A192254 0-sequence of reduction of (n^2) by x^2 -> x+1.
%C A192254 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192254 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
%t A192254 c[n_] := n^2;  (* A000290 *)
%t A192254 Table[c[n], {n, 1, 15}]
%t A192254 q[x_] := x + 1;
%t A192254 p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
%t A192254 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192254    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192254 t = Table[
%t A192254   Last[Most[
%t A192254     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192254    30}]
%t A192254 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192254 *)
%t A192254 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192255 *)
%t A192254 (* _Peter J. C. Moses_, Jun 20 2011 *)
%K A192254 nonn,changed
%O A192254 1,3
%A A192254 _Clark Kimberling_, Jun 27 2011