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A192299 0-sequence of reduction of (n^2+n+1) by x^2 -> x+1. 1

%I

%S 1,1,8,21,63,156,371,827,1776,3687,7461,14776,28749,55101,104264,

%T 195121,361651,664660,1212431,2196935,3957136,7089331,12638953,

%U 22433136,39655993,69841561,122584136,214478637,374166471,650979852

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

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

%F Empirical G.f.: x*(1-3*x+7*x^2-3*x^3+3*x^4-x^5)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 10 2012]

%t c[n_] := n^2 + n + 1;(* central polygonal numbers starting at 3 *)

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

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

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

%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 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192300 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192142 *)

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

%Y Cf. A192232, A192142.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 27 2011

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Last modified October 23 17:37 EDT 2021. Contains 348215 sequences. (Running on oeis4.)