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A192307 0-sequence of reduction of (3n) by x^2 -> x+1.

%I A192307 #19 Aug 23 2025 23:12:06
%S A192307 3,3,12,24,54,108,213,405,756,1386,2508,4488,7959,14007,24492,42588,
%T A192307 73698,126996,218025,373065,636468,1082958,1838232,3113424,5262699,
%U A192307 8879403,14956428,25153440,42241806,70844796,118668093,198543933,331824564,554012082,924092772
%N A192307 0-sequence of reduction of (3n) by x^2 -> x+1.
%C A192307 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192307 a(n) = 3*A190062(n).
%F A192307 G.f.: 3*x*(1-2*x+2*x^2)/(1-x)/(1-x-x^2)^2. - _Colin Barker_, Feb 11 2012
%t A192307 c[n_] := 3 n;
%t A192307 Table[c[n], {n, 1, 15}]
%t A192307 q[x_] := x + 1;
%t A192307 p[0, x_] := 3; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
%t A192307 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192307 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 40}]
%t A192307 Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192307 *)
%t A192307 Table[Coefficient[Part[t, n]/3, x, 0], {n, 1, 40}]  (* A190062 *)
%t A192307 Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192308 *)
%t A192307 Table[Coefficient[Part[t, n]/3, x, 1], {n, 1, 40}]  (* A122491 *)
%t A192307 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192307 Cf. A192232, A192307.
%K A192307 nonn,easy,changed
%O A192307 1,1
%A A192307 _Clark Kimberling_, Jun 27 2011
%E A192307 More terms from _Jason Yuen_, Aug 23 2025