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A192309 0-sequence of reduction of (3n-1) by x^2 -> x+1.

%I A192309 #13 Aug 23 2025 23:12:16
%S A192309 2,2,10,21,49,100,200,384,722,1331,2419,4344,7726,13630,23882,41601,
%T A192309 72101,124412,213844,366300,625522,1065247,1809575,3067056,5187674,
%U A192309 8758010,14760010,24835629,41727577,70012756,117321824,196365624,328299986,548309195,914865307
%N A192309 0-sequence of reduction of (3n-1) by x^2 -> x+1.
%C A192309 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192309 Empirical g.f.: x*(2-4*x+6*x^2-x^3)/(1-3*x+x^2+3*x^3-x^4-x^5). - _Colin Barker_, Feb 09 2012
%t A192309 c[n_] := 3 n - 1;
%t A192309 Table[c[n], {n, 1, 15}]
%t A192309 q[x_] := x + 1;
%t A192309 p[0, x_] := 2; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
%t A192309 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192309 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 40}]
%t A192309 Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192309 *)
%t A192309 Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192310 *)
%t A192309 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192309 Cf. A192232, A192310.
%K A192309 nonn
%O A192309 1,1
%A A192309 _Clark Kimberling_, Jun 27 2011
%E A192309 More terms from _Jason Yuen_, Aug 23 2025