cp's OEIS Frontend

A192305 0-sequence of reduction of (2n) by x^2 -> x+1.

%I A192305 #16 Aug 23 2025 08:02:24
%S A192305 2,2,8,16,36,72,142,270,504,924,1672,2992,5306,9338,16328,28392,49132,
%T A192305 84664,145350,248710,424312,721972,1225488,2075616,3508466,5919602,
%U A192305 9970952,16768960,28161204,47229864,79112062,132362622,221216376,369341388,616061848,1026669712,1709502122,2844208874,4728518600,7855572120
%N A192305 0-sequence of reduction of (2n) by x^2 -> x+1.
%C A192305 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192305 a(n) = 2*A190062(n).
%F A192305 G.f.: 2*x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2). - _Colin Barker_, Feb 11 2012
%t A192305 c[n_] := 2 n; (* even numbers, A005843 *)
%t A192305 Table[c[n], {n, 1, 15}]
%t A192305 q[x_] := x + 1;
%t A192305 p[0, x_] := 2; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
%t A192305 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192305    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192305 t = Table[
%t A192305   Last[Most[
%t A192305     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192305    40}]
%t A192305 Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192305 *)
%t A192305 Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 40}]  (* A190062 *)
%t A192305 Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192306 *)
%t A192305 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}]  (* A122491 *)
%t A192305 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192305 Cf. A192232, A192306, A192304.
%K A192305 nonn,easy,changed
%O A192305 1,1
%A A192305 _Clark Kimberling_, Jun 27 2011