A192309 0-sequence of reduction of (3n-1) by x^2 -> x+1.

2, 2, 10, 21, 49, 100, 200, 384, 722, 1331, 2419, 4344, 7726, 13630, 23882, 41601, 72101, 124412, 213844, 366300, 625522, 1065247, 1809575, 3067056, 5187674, 8758010, 14760010, 24835629, 41727577, 70012756, 117321824, 196365624, 328299986, 548309195, 914865307

Offset: 1

Author

Clark Kimberling, Jun 27 2011

Keywords

• nonn

Comments

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

Crossrefs

Cf. A192232, A192310.

Programs

Mathematica

c[n_] := 3 n - 1;
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 2; 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}]  (* A192309 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A192310 *)
(* Peter J. C. Moses, Jun 20 2011 *)

Formula

Empirical g.f.: x*(2-4*x+6*x^2-x^3)/(1-3*x+x^2+3*x^3-x^4-x^5). - Colin Barker, Feb 09 2012

Extensions

More terms from Jason Yuen, Aug 23 2025