A289920 p-INVERT of (1,0,0,1,0,0,1,0,0,...), where p(S) = 1 - S - S^2.

1, 2, 3, 6, 12, 22, 42, 80, 151, 287, 544, 1031, 1956, 3708, 7031, 13333, 25280, 47936, 90895, 172350, 326806, 619677, 1175008, 2228011, 4224672, 8010672, 15189552, 28801880, 54613096, 103555397, 196358029, 372327066, 705993241, 1338679088, 2538355336

Offset: 0

Author

Clark Kimberling, Sep 14 2017

Keywords

• nonn
• easy

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A291728 for a guide to related sequences.

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 1, 2, -1, 0, -1)

Crossrefs

Cf. A079978, A289918.

Programs

Mathematica

z = 60; s = x/(x - x^3); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A079978  *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A289920 *)

Formula

G.f.: -((-1 - x + x^3)/(1 - x - x^2 - 2 x^3 + x^4 + x^6)).

a(n) = a(n-1) + a(n-2) + 2*a(n-3) - a(n-4) - a(n-6) for n >= 7.