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%I #22 Jan 05 2023 03:20:36
%S 1,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,
%T 1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,
%U 0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,0,1,1,1,2,1
%N Expansion of (1 + x^5) / ((1 - x) * (1 + x^4)) in powers of x.
%H Antti Karttunen, <a href="/A257179/b257179.txt">Table of n, a(n) for n = 0..65537</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,-1,1).
%F Euler transform of length 10 sequence [1, 0, 0, -1, 1, 0, 0, 1, 0, -1].
%F Moebius transform is length 8 sequence [1, 0, 0, -1, 0, 0, 0, 2].
%F a(n) is multiplicative with a(2) = 1, a(4) = 0, a(2^e) = 2 if e>2, a(p^e) = 1 if p>2 and a(0) = 1.
%F G.f.: (1 + x^5) / ((1 - x) * (1 + x^4)).
%F G.f.: (1 - x^4) * (1 - x^10) / ((1 - x) * (1 - x^5) * (1 - x^8)).
%F G.f.: -1 + 1 / (1 - x) + 1 / (1 + x^4).
%F a(n) = a(-n) for all n in Z. a(n+8) = a(n) unless n=0 or n=-8. a(8*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1. a(8*n + 4) = 0.
%F a(n) = A259042(n+4) unless n = 0.
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - _Amiram Eldar_, Nov 20 2022
%F Dirichlet g.f.: zeta(s)*(1-1/4^s+2/8^s). - _Amiram Eldar_, Jan 05 2023
%e G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + x^11 + ...
%t a[ n_] := Boole[n != 0] - Boole[Mod[n, 4] == 0] + 2 Boole[Mod[n, 8] == 0];
%t a[ n_] := -Boole[n == 0] + {1, 1, 1, 0, 1, 1, 1, 2}[[Mod[n, 8, 1]]];
%t a[ n_] := SeriesCoefficient[ (1 + x^5) / ((1 - x) * (1 + x^4)), {x, 0, Abs@n}];
%o (PARI) {a(n) = (n != 0) - (n%4 == 0) + 2*(n%8 == 0)};
%o (PARI) {a(n) = -(n==0) + [2, 1, 1, 1, 0, 1, 1, 1][n%8 + 1]};
%o (PARI) {a(n) = polcoeff( (1 + x^5) / ((1 - x) * (1 + x^4)) + x * O(x^abs(n)), abs(n))};
%Y Cf. A010877, A259042.
%K nonn,mult,easy
%O 0,9
%A _Michael Somos_, Apr 17 2015
%E More terms from _Antti Karttunen_, Jul 29 2018
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