A289782 p-INVERT of the Lucas numbers (A000032), where p(S) = 1 - S - S^2.

2, 9, 35, 146, 593, 2428, 9911, 40495, 165399, 675637, 2759792, 11273144, 46048100, 188095781, 768327108, 3138436438, 12819777601, 52365789305, 213901984464, 873739509697, 3569021260182, 14578615958179, 59550231769665, 243248749683441, 993614171826023

Offset: 0

Author

Clark Kimberling, Aug 10 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 A289780 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 (4, 2, -7, 1)

Crossrefs

Cf. A000032, A289780.

Programs

Mathematica

z = 60; s = (2 - x) x/(1 - x - x^2); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000032 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289782 *)

Formula

G.f.: (-2 - x + 5 x^2 - 2 x^3)/(-1 + 4 x + 2 x^2 - 7 x^3 + x^4).

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