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

%I A242668 #34 Mar 31 2023 14:34:10
%S A242668 1,8,48,256,1279,6132,28576,130432,585985,2599952,11419808,49743104,
%T A242668 215163647,925163500,3957669648,16854677312,71498512897,302248757272,
%U A242668 1273756836176,5353050574336,22440215412223,93856659402724,391745066819136,1631995960879872
%N A242668 Expansion of 1/(1 - 8*x + 16*x^2 + x^4 - 4*x^5).
%C A242668 Subsequence of A047538.
%C A242668 a(n) is divisible by 4^(n mod 4).
%D A242668 C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), 229-230 (example 9.43).
%H A242668 Bruno Berselli, <a href="/A242668/b242668.txt">Table of n, a(n) for n = 0..200</a>
%H A242668 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16,0,-1,4).
%F A242668 G.f.: 1/((1 - 4*x)*(1 - 4*x + x^4)).
%F A242668 a(n) = 8*a(n-1) - 16*a(n-2) - a(n-4) + 4*a(n-5) for n>4.
%F A242668 a(n) = 4*a(n-1) - a(n-4) + 4^n for n>3 (see References, p. 229).
%F A242668 Trisections:
%F A242668 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 A242668 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 A242668 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 A242668 a(n) ~ 4^(4+n). - _Stefano Spezia_, Mar 29 2023
%t A242668 CoefficientList[Series[1/((1 - 4 x) (1 - 4 x + x^4)), {x, 0, 30}], x]
%t A242668 LinearRecurrence[{8,-16,0,-1,4},{1,8,48,256,1279},40] (* _Harvey P. Dale_, Aug 10 2021 *)
%o A242668 (PARI) Vec(1/(1-8*x+16*x^2+x^4-4*x^5)+O(x^30))
%o A242668 (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 A242668 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-8*x+16*x^2+x^4-4*x^5)));
%o A242668 (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 A242668 Cf. A047538.
%K A242668 nonn,easy
%O A242668 0,2
%A A242668 _Bruno Berselli_, May 20 2014