A242668 - OEIS

%I #34 Mar 31 2023 14:34:10

%S 1,8,48,256,1279,6132,28576,130432,585985,2599952,11419808,49743104,

%T 215163647,925163500,3957669648,16854677312,71498512897,302248757272,

%U 1273756836176,5353050574336,22440215412223,93856659402724,391745066819136,1631995960879872

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

%C Subsequence of A047538.

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

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

%H Bruno Berselli, <a href="/A242668/b242668.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,0,-1,4).

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

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

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

%F Trisections:

%F 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));

%F 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));

%F 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)).

%F a(n) ~ 4^(4+n). - _Stefano Spezia_, Mar 29 2023

%t CoefficientList[Series[1/((1 - 4 x) (1 - 4 x + x^4)), {x, 0, 30}], x]

%t LinearRecurrence[{8,-16,0,-1,4},{1,8,48,256,1279},40] (* _Harvey P. Dale_, Aug 10 2021 *)

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

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

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

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

%Y Cf. A047538.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, May 20 2014