A242668 Expansion of 1/(1 - 8*x + 16*x^2 + x^4 - 4*x^5).

1, 8, 48, 256, 1279, 6132, 28576, 130432, 585985, 2599952, 11419808, 49743104, 215163647, 925163500, 3957669648, 16854677312, 71498512897, 302248757272, 1273756836176, 5353050574336, 22440215412223, 93856659402724, 391745066819136, 1631995960879872

Offset: 0

Author

Bruno Berselli, May 20 2014

Keywords

• nonn
• easy

Comments

Subsequence of A047538.

a(n) is divisible by 4^(n mod 4).

References

• C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), 229-230 (example 9.43).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (8,-16,0,-1,4).

Crossrefs

Cf. A047538.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-8*x+16*x^2+x^4-4*x^5)));

Mathematica

CoefficientList[Series[1/((1 - 4 x) (1 - 4 x + x^4)), {x, 0, 30}], x]
LinearRecurrence[{8,-16,0,-1,4},{1,8,48,256,1279},40] (* Harvey P. Dale, Aug 10 2021 *)

Maxima

makelist(coeff(taylor(1/(1-8*x+16*x^2+x^4-4*x^5), x, 0, n), x, n), n, 0, 30);

PARI

Vec(1/(1-8*x+16*x^2+x^4-4*x^5)+O(x^30))

Sage

m = 30; L. = PowerSeriesRing(ZZ, m); f = 1/(1-8*x+16*x^2+x^4-4*x^5); print(f.coefficients())

Formula

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

a(n) = 8*a(n-1) - 16*a(n-2) - a(n-4) + 4*a(n-5) for n>4.

a(n) = 4*a(n-1) - a(n-4) + 4^n for n>3 (see References, p. 229).

Trisections:

a(3k): 1, 256, 28576, 2599952, 215163647, 16854677312, 1273756836176, ... has g.f. (1+128*x-48*x^2+4*x^3)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4));

a(3k+1): 8, 1279, 130432, 11419808, 925163500, 71498512897, ... has g.f. (8+255*x-128*x^2+16*x^3)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4));

a(3k+2): 48, 6132, 585985, 49743104, 3957669648, 302248757272, ... has g.f. (48-12*x+x^2)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4)).

a(n) ~ 4^(4+n). - Stefano Spezia, Mar 29 2023