A192305 0-sequence of reduction of (2n) by x^2 -> x+1.
2, 2, 8, 16, 36, 72, 142, 270, 504, 924, 1672, 2992, 5306, 9338, 16328, 28392, 49132, 84664, 145350, 248710, 424312, 721972, 1225488, 2075616, 3508466, 5919602, 9970952, 16768960, 28161204, 47229864, 79112062, 132362622, 221216376, 369341388, 616061848, 1026669712, 1709502122, 2844208874, 4728518600, 7855572120
Offset: 1
Programs
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Mathematica
c[n_] := 2 n; (* even numbers, A005843 *) Table[c[n], {n, 1, 15}] q[x_] := x + 1; p[0, x_] := 2; 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, 40}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}] (* A192305 *) Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 40}] (* A190062 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}] (* A192306 *) Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 40}] (* A122491 *) (* Peter J. C. Moses, Jun 20 2011 *)
Formula
a(n) = 2*A190062(n).
G.f.: 2*x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2). - Colin Barker, Feb 11 2012
Comments