A292327 p-INVERT of the Fibonacci sequence (A000045), where p(S) = (1 - S)^2.

2, 5, 14, 38, 102, 271, 714, 1868, 4858, 12569, 32374, 83058, 212350, 541219, 1375570, 3487384, 8821170, 22266413, 56098206, 141087934, 354268502, 888238903, 2223968666, 5561234916, 13889778218, 34652529473, 86361653126, 215021205770, 534861620718

Offset

0,1

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 (4,-2,-4,-1).

Formula

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

a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4) for n >= 5.

a(n) = A006645(n+1) +2*A000129(n+1). - R. J. Mathar, Jul 08 2022

Mathematica

z = 60; s = x/(1 - x - x^2); p = (1 - s)^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000045 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292327 *)

Crossrefs

Cf. A000045, A292328.

Cf. A006645, A000129.

Sequence in context: A148313 A228952 A172259 * A084085 A052985 A052945

Adjacent sequences: A292324 A292325 A292326 * A292328 A292329 A292330

Keyword

nonn,easy

Author

Clark Kimberling, Sep 15 2017

Status

approved