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A280057 a(n) = (-1)^n * 4*n/3 if n = 3*k and n!=0, otherwise a(n) = (-1)^n * n except a(0) = 1.

%I A280057 #29 Sep 08 2022 08:46:18
%S A280057 1,-1,2,-4,4,-5,8,-7,8,-12,10,-11,16,-13,14,-20,16,-17,24,-19,20,-28,
%T A280057 22,-23,32,-25,26,-36,28,-29,40,-31,32,-44,34,-35,48,-37,38,-52,40,
%U A280057 -41,56,-43,44,-60,46,-47,64,-49,50,-68,52,-53,72,-55,56,-76,58,-59
%N A280057 a(n) = (-1)^n * 4*n/3 if n = 3*k and n!=0, otherwise a(n) = (-1)^n * n except a(0) = 1.
%C A280057 This is a divisibility sequence.
%H A280057 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,-1).
%F A280057 Euler transform of length 10 sequence [-1, 2, -2, -1, 1, 2, 0, 0, 0, -1]. - _Michael Somos_, Dec 29 2016
%F A280057 a(n) = -b(n) where b() is multiplicative with b(2^e) = -(2^e) if e>0, b(3^e) = 4*3^(e-1) if e>0, a(p^e) = p^e otherwise.
%F A280057 G.f.: 1 - x / (1 + x)^2 - x^3 / (1 + x^3)^2.
%F A280057 G.f.: (1 + x^2) * (1 + x^5) / ((1 + x) * (1 + x^3)^2).
%F A280057 G.f.: (1 - x) * (1 - x^3)^2 * (1 - x^4) * (1 - x^10) / ((1 - x^2)^2 * (1 - x^5) * (1 - x^6)^2).
%F A280057 a(n) = (-1)^n * A257174(n).
%e A280057 G.f. = 1 - x + 2*x^2 - 4*x^3 + 4*x^4 - 5*x^5 + 8*x^6 - 7*x^7 + 8*x^8 + ...
%t A280057 a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n n If[ Mod[n, 3] > 0, 1, 4/3]];
%t A280057 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (1 + x^2) (1 + x^5) / ((1 + x) (1 + x^3)^2), {x, 0, n}]];
%o A280057 (PARI) {a(n) = if( n<1, n==0, (-1)^n * n * if( n%3, 1, 4/3))};
%o A280057 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x^2) * (1 + x^5) / ((1 + x) * (1 + x^3)^2) + x * O(x^n), n))};
%o A280057 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - x) * (1 - x^3)^2 * (1 - x^4) * (1 - x^10) / ((1 - x^2)^2 * (1 - x^5) * (1 - x^6)^2) + x * O(x^n), n))};
%o A280057 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x^2)*(1+x^5)/((1+x)*(1+x^3)^2))); // _G. C. Greubel_, Jul 29 2018
%Y A280057 Cf. A257174.
%K A280057 sign
%O A280057 0,3
%A A280057 _Michael Somos_, Dec 29 2016