A192305 - OEIS

%I #10 Dec 04 2016 19:46:25

%S 2,2,8,16,36,72,142,270,504,924,1672,2992,5306,9338,16328,28392,49132,

%T 84664,145350,248710,424312,721972,1225488,2075616,3508466,5919602,

%U 9970952,16768960,28161204,47229864,79112062,132362622,221216376,369341388,616061848,1026669712,1709502122,2844208874,4728518600,7855572120

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

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

%F a(n) = 2*A190062(n).

%F G.f.: 2*x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2). [Colin Barker, Feb 11 2012]

%t c[n_] := 2 n; (* even numbers, A005843 *)

%t Table[c[n], {n, 1, 15}]

%t q[x_] := x + 1;

%t p[0, x_] := 2; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t    x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[

%t   Last[Most[

%t     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,

%t    40}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192305  *)

%t Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 40}]  (* A190062 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192306 *)

%t Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}]  (* A122491 *)

%t (* by _Peter J. C. Moses_, Jun 20 2011 *)

%Y Cf. A192232, A192306, A192304.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 27 2011