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Expansion of Sum_{k>0} x^k / (1 + x^(3*k)).
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%I #25 Jul 02 2023 10:03:16

%S 1,1,1,0,1,1,2,0,1,0,1,0,2,2,1,-1,1,1,2,-1,2,0,1,0,2,2,1,0,1,0,2,-1,1,

%T 0,2,0,2,2,2,-2,1,2,2,-1,1,0,1,-1,3,1,1,0,1,1,2,0,2,0,1,-1,2,2,2,-2,2,

%U 0,2,-1,1,0,1,0,2,2,2,0,2,2,2,-3,1,0,1,0,2,2,1,-2,1,0,4,-1,2,0

%N Expansion of Sum_{k>0} x^k / (1 + x^(3*k)).

%H Seiichi Manyama, <a href="/A364011/b364011.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>0} (-1)^(k-1) * x^(3*k-2) / (1 - x^(3*k-2)).

%F a(n) = -Sum_{d|n, n/d==1 (mod 3)} (-1)^(n/d) = -Sum_{d|n, d==1 (mod 3)} (-1)^d.

%t a[n_] := -DivisorSum[n, (-1)^(n/#) &, Mod[n/#, 3] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Jul 01 2023 *)

%o (PARI) a(n) = -sumdiv(n, d, (d%3==1)*(-1)^d);

%Y Cf. A364012, A364013.

%Y Cf. A002654, A363037.

%K sign

%O 1,7

%A _Seiichi Manyama_, Jul 01 2023