A192299 0-sequence of reduction of (n^2+n+1) by x^2 -> x+1.

1, 1, 8, 21, 63, 156, 371, 827, 1776, 3687, 7461, 14776, 28749, 55101, 104264, 195121, 361651, 664660, 1212431, 2196935, 3957136, 7089331, 12638953, 22433136, 39655993, 69841561, 122584136, 214478637, 374166471, 650979852

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+7*x^2-3*x^3+3*x^4-x^5)/(1-x)/(1-x-x^2)^3. [Colin Barker, Feb 10 2012]

Mathematica

c[n_] := n^2 + n + 1; (* central polygonal numbers starting at 3 *)
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]
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}]  (* A192300 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192142 *)
(* by Peter J. C. Moses, Jun 20 2011 *)

Crossrefs

Cf. A192232, A192142.

Sequence in context: A227653 A296198 A301538 * A080144 A241522 A096018

Adjacent sequences: A192296 A192297 A192298 * A192300 A192301 A192302

Keyword

nonn

Author

Clark Kimberling, Jun 27 2011

Status

approved