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A192304 0-sequence of reduction of (2n-1) by x^2 -> x+1.

%I A192304 #14 Aug 23 2025 08:03:11
%S A192304 1,1,6,13,31,64,129,249,470,869,1583,2848,5073,8961,15718,27405,47535,
%T A192304 82080,141169,241945,413366,704261,1196831,2029248,3433441,5798209,
%U A192304 9774534,16451149,27646975,46397824
%N A192304 0-sequence of reduction of (2n-1) by x^2 -> x+1.
%C A192304 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192304 Empirical g.f.: x*(1-2*x+4*x^2-x^3)/(1-3*x+x^2+3*x^3-x^4-x^5). - _Colin Barker_, Feb 08 2012
%t A192304 c[n_] := 2 n - 1; (* odd numbers, A005408 *)
%t A192304 Table[c[n], {n, 1, 15}]
%t A192304 q[x_] := x + 1;
%t A192304 p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
%t A192304 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192304    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192304 t = Table[
%t A192304   Last[Most[
%t A192304     FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192304    30}]
%t A192304 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192304 *)
%t A192304 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A178525 *)
%t A192304 (* _Peter J. C. Moses_, Jun 20 2011 *)
%Y A192304 Cf. A192232, A178525.
%K A192304 nonn,more,changed
%O A192304 1,3
%A A192304 _Clark Kimberling_, Jun 27 2011