A292328 -id:A292328 - cp's OEIS frontend

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

Clark Kimberling, Sep 15 2017

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).

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000045, A292328.

Cf. A006645, A000129.

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 *)

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

A292329 p-INVERT of the Fibonacci sequence (A000045), where p(S) = 1 - S^3.

Original entry on oeis.org

0, 0, 1, 3, 9, 23, 57, 138, 332, 798, 1920, 4626, 11157, 26925, 64997, 156921, 378861, 914692, 2208324, 5331444, 12871324, 31074180, 75019701, 181113471, 437246349, 1055605659, 2548456957, 6152518758, 14853493752, 35859505946, 86572506132, 209004519918

Offset: 0

Clark Kimberling, Sep 15 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, 0, -4, 0, 3, 1)

Cf. A000045, A292328.

z = 60; s = x/(1 - x - x^2); p = 1 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000045 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A292329 *)

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

a(n) = 3*a(n-1) - 4*a(n-3) + 3*a(n-5) + a(n-6) for n >= 7.

cp's OEIS Frontend

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

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