A289783 p-INVERT of the (3^n), where p(S) = 1 - S - S^2.

1, 5, 24, 113, 527, 2446, 11325, 52369, 242008, 1117997, 5163891, 23849270, 110142089, 508652653, 2349005592, 10847859961, 50095958215, 231345247934, 1068361195173, 4933730638937, 22784141325656, 105217952251285, 485900111176779, 2243903303473318

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).

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-11).

Crossrefs

Cf. A000244, A289780.

Programs

Mathematica

z = 60; s = x/(1 - 3*x); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000244 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289783 *)

PARI

Vec(x*(1 - 2*x) / (1 - 7*x + 11*x^2) + O(x^30)) \\ Colin Barker, Aug 11 2017

Formula

G.f.: (1 - 2 x)/(1 - 7 x + 11 x^2).

a(n) = 7*a(n-1) - 11*a(n-2).

a(n) = (2^(-n-1)*((7-sqrt(5))^(n+1)*(-4+sqrt(5)) + (4+sqrt(5))*(7+sqrt(5))^(n+1))) / (11*sqrt(5)). - Colin Barker, Aug 11 2017

a(n) = A099453(n) - 2*A099453(n-1). - R. J. Mathar, Jul 08 2022

E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Aug 05 2025