A256096 Expansion of (4+3*x)/(1+3*x).

4, -9, 27, -81, 243, -729, 2187, -6561, 19683, -59049, 177147, -531441, 1594323, -4782969, 14348907, -43046721, 129140163, -387420489, 1162261467, -3486784401, 10460353203, -31381059609, 94143178827, -282429536481, 847288609443, -2541865828329, 7625597484987

Offset: 0

Author

Wolfdieter Lang, Mar 23 2015

Keywords

• sign
• easy

Comments

This is the Z-sequence of the Riordan triangle ((1+2*x)/(1-x)^2, x/(1-x)). See A135857.

The expansion can be seen as a special case of the family of generating functions 1+1/(x+1/k) for k>=1 which relates this sequence to A054977 and A198633 (neglecting questions of sign). - Peter Luschny, Mar 24 2015

Links

• Winston de Greef, Table of n, a(n) for n = 0..2080
• Index entries for linear recurrences with constant coefficients, signature (-3).

Crossrefs

Cf. A135857, A054977, A198633.

Programs

Mathematica

Join[{4},NestList[-3#&,-9,30]] (* Harvey P. Dale, Jan 27 2023 *)

PARI

a(n)=if(!n, 4, (-1)^n*3^(n+1)) \\ Winston de Greef, Mar 19 2023

Formula

O.g.f.: (4+3*x)/(1+3*x).

a(0) = 4; for n >= 1, a(n) = (-1)^n*3^(n+1).

a(0) = 4, a(1) = -9; for n >= 2, a(n) = -3*a(n-1).

E.g.f.: 1 + 3*exp(-3*x). - Alejandro J. Becerra Jr., Jan 28 2021