A192307 0-sequence of reduction of (3n) by x^2 -> x+1.

3, 3, 12, 24, 54, 108, 213, 405, 756, 1386, 2508, 4488, 7959, 14007, 24492, 42588, 73698, 126996, 218025, 373065, 636468, 1082958, 1838232, 3113424, 5262699, 8879403, 14956428, 25153440, 42241806, 70844796, 118668093, 198543933, 331824564, 554012082, 924092772

Offset: 1

Author

Clark Kimberling, Jun 27 2011

Keywords

• nonn
• easy
• changed

Comments

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

Crossrefs

Cf. A192232, A192307.

Programs

Mathematica

c[n_] := 3 n;
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 3; 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, 40}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192307 *)
Table[Coefficient[Part[t, n]/3, x, 0], {n, 1, 40}]  (* A190062 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192308 *)
Table[Coefficient[Part[t, n]/3, x, 1], {n, 1, 40}]  (* A122491 *)
(* Peter J. C. Moses, Jun 20 2011 *)

Formula

a(n) = 3*A190062(n).

G.f.: 3*x*(1-2*x+2*x^2)/(1-x)/(1-x-x^2)^2. - Colin Barker, Feb 11 2012

Extensions

More terms from Jason Yuen, Aug 23 2025