A291222 p-INVERT of (0,1,0,1,0,1,...), where p(S) = 1 - S^2 - S^3.

0, 1, 1, 3, 5, 9, 19, 30, 66, 106, 223, 379, 753, 1345, 2565, 4723, 8816, 16456, 30480, 57093, 105677, 197751, 366697, 684765, 1272311, 2371846, 4412898, 8218386, 15300891, 28483823, 53042669, 98734485, 183863833, 342263703, 637320032, 1186464528, 2209131168

Offset: 0

Author

Clark Kimberling, Aug 24 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 A291219 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 (0, 4, 1, -4, 0, 1)

Crossrefs

Cf. A000035, A291219.

Programs

Mathematica

z = 60; s = x/(1 - x^2); p = 1 - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000035 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291222 *)

PARI

concat(0, Vec(x*(1 + x - x^2) / (1 - 4*x^2 - x^3 + 4*x^4 - x^6) + O(x^40))) \\ Colin Barker, Aug 25 2017

Formula

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

G.f.: x*(1 + x - x^2) / (1 - 4*x^2 - x^3 + 4*x^4 - x^6). - Colin Barker, Aug 25 2017