A192145 0-sequence of reduction of pentagonal numbers sequence by x^2 -> x+1.

1, 1, 13, 35, 105, 258, 608, 1344, 2865, 5910, 11894, 23444, 45427, 86755, 163645, 305397, 564647, 1035446, 1885050, 3409610, 6131441, 10968416, 19528188, 34617960, 61125685, 107540053, 188567053, 329625719, 574558965, 998836650

Offset

1,3

Comments

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

Links

Table of n, a(n) for n=1..30.

Formula

Empirical G.f.: x*(1-3*x+12*x^2-9*x^3+4*x^4)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 11 2012]

Mathematica

Remove["Global`*"];
c[n_] := n (3 n - 1)/2; (* pentagonal numbers, A000326 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[
  Last[Most[
    FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
   30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192145 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192146 *)
(* by Peter J. C. Moses, Jun 20 2011 *)

Crossrefs

Cf. A192232, A192146.

Sequence in context: A228024 A243383 A242578 * A221592 A135172 A183309

Adjacent sequences: A192142 A192143 A192144 * A192146 A192147 A192148

Keyword

nonn

Author

Clark Kimberling, Jun 27 2011

Status

approved