|
%I #29 Dec 29 2022 06:30:34
%S 0,1,-2,2,-4,5,-4,7,-8,6,-10,11,-8,13,-14,10,-16,17,-12,19,-20,14,-22,
%T 23,-16,25,-26,18,-28,29,-20,31,-32,22,-34,35,-24,37,-38,26,-40,41,
%U -28,43,-44,30,-46,47,-32,49,-50,34,-52,53,-36,55,-56,38,-58,59
%N a(n) = -(-1)^n * 2 * n / 3 if n divisible by 3, a(n) = -(-1)^n * n otherwise.
%H G. C. Greubel, <a href="/A255368/b255368.txt">Table of n, a(n) for n = 0..2500</a>
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.txt">Rational function multiplicative coefficients</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,-1).
%F Euler transform of length 6 sequence [-2, 1, -2, -1, 0, 2].
%F a(n) is multiplicative with a(2^e) = -(2^e) if e>0, a(3^e) = 2 * 3^(e-1) if e>0, otherwise a(p^e) = p^e.
%F G.f.: f(x) - f(x^3) where f(x) := x / (1 + x)^2.
%F G.f.: x * (1 - x)^2 * (1 + x^2) / (1 + x^3)^2.
%F G.f.: x * (1 - x)^2 * (1 - x^3)^2 * (1 - x^4) / ((1 - x^2) * (1 - x^6)^2).
%F a(n) = -a(-n) = -(-1)^n * A186101(n) for all n in Z.
%F Dirichlet g.f.: zeta(s-1)*(2^s-4)*(3^s-1)/6^s. - _Amiram Eldar_, Dec 29 2022
%e G.f. = x - 2*x^2 + 2*x^3 - 4*x^4 + 5*x^5 - 4*x^6 + 7*x^7 - 8*x^8 + 6*x^9 + ...
%t a[ n_] := -(-1)^n If[ Divisible[ n, 3], 2 n/3, n];
%t a[ n_] := n {1, -1, 2/3, -1, 1, -2/3}[[Mod[n, 6, 1]]];
%t CoefficientList[Series[x*(1-x)^2*(1+x^2)/(1+x^3)^2, {x,0,60}], x] (* _G. C. Greubel_, Aug 02 2018 *)
%o (PARI) {a(n) = -(-1)^n * if( n%3, n, 2*n/3)};
%o (PARI) my(x='x+O('x^60)); concat([0], Vec(x*(1-x)^2*(1+x^2)/(1+x^3)^2)) \\ _G. C. Greubel_, Aug 02 2018
%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1-x)^2*(1+x^2)/(1+x^3)^2)); // _G. C. Greubel_, Aug 02 2018
%Y Cf. A186101.
%K sign,mult,easy
%O 0,3
%A _Michael Somos_, May 04 2015
|
