A192304 - OEIS

%I #8 Dec 04 2016 19:46:25

%S 1,1,6,13,31,64,129,249,470,869,1583,2848,5073,8961,15718,27405,47535,

%T 82080,141169,241945,413366,704261,1196831,2029248,3433441,5798209,

%U 9774534,16451149,27646975,46397824

%N 0-sequence of reduction of (2n-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*(1-2*x+4*x^2-x^3)/(1-3*x+x^2+3*x^3-x^4-x^5). [Colin Barker, Feb 08 2012]

%t c[n_] := 2 n - 1; (* odd numbers, A005408 *)

%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 + 1]

%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}]  (* A192304  *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A178525 *)

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

%Y Cf. A192232, A178525.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 27 2011