A291001 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 8*S^2.

0, 8, 16, 88, 288, 1192, 4400, 17144, 65088, 250184, 955984, 3663256, 14018400, 53679592, 205487984, 786733112, 3011882112, 11530896008, 44144966800, 169006205656, 647027178912, 2477097797416, 9483385847216, 36306456276344, 138996613483200, 532138420900808

Offset: 0

Author

Clark Kimberling, Aug 22 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 A291000 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 (2,7).

Crossrefs

Cf. A000012, A015519, A033453, A289780, A291000.

Programs

Magma

[n le 2 select 8*(n-1) else 2*Self(n-1) +7*Self(n-2): n in [1..41]]; // G. C. Greubel, Apr 25 2023

Mathematica

z = 60; s = x/(1 - x); p = 1 - s^8;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291001 *)
LinearRecurrence[{2,7}, {0,8}, 41] (* G. C. Greubel, Apr 25 2023 *)

SageMath

A291001=BinaryRecurrenceSequence(2,7,0,8)
[A291001(n) for n in range(41)] # G. C. Greubel, Apr 25 2023

Formula

G.f.: 8*x/(1 - 2*x - 7*x^2).

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

a(n) = 8*A015519(n).

a(n) = sqrt(2)*((1+2*sqrt(2))^n - (1-2*sqrt(2))^n). - Colin Barker, Aug 23 2017