A192143 - OEIS

%I #9 Dec 04 2016 19:46:25

%S 1,1,16,44,134,332,787,1747,3736,7726,15580,30760,59685,114117,215472,

%T 402464,744674,1366484,2489175,4504695,8104536,14504226,25833336,

%U 45811344,80916169,142400137,249760912,436706132,761385086,1323910556

%N 0-sequence of reduction of hexagonal numbers sequence 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+15*x^2-12*x^3+6*x^4)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 11 2012]

%t c[n_] := n (2 n - 1); (* hexagonal numbers, A000384 *)

%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}]  (* A192143 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192144 *)

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

%Y Cf. A192232, A192144.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 27 2011