|
%I #16 Sep 08 2022 08:45:54
%S 1,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,
%T -4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,
%U 2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2,-1,-4,-1,2,2,2
%N Expansion of (1 - x^2)^2 * (1 + x)^2 / (1 - x^6) in powers of x.
%H G. C. Greubel, <a href="/A181982/b181982.txt">Table of n, a(n) for n = 0..5000</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, -1, 0, -1).
%F Euler transform of length 6 sequence [2, -4, 0, 0, 0, 1].
%F a(n) = 2 * b(n) where b(n) is multiplicative with b(2^e) = -1/2 if e>0, b(3^e) = -2 if e>0, b(p^e) = 1 if p>3.
%F Expansion of 1 + 2*x / (1 - x) - 3*x^2 / (1 - x^2) - 6*x^3 / (1 - x^3) + 9*x^6 / (1 - x^6) in powers of x.
%F G.f.: (1 - x) * (1 + x)^3 / (1 + x^2 + x^4) = (1 - x^2)^4 / ((1 - x)^2 * (1 - x^6)).
%F a(-n) = a(n). a(n - 6) = a(n) unless n=0. a(n) = -a(n-2) - a(n-4) unless n=0, 2, 4.
%e G.f. = 1 + 2*x - x^2 - 4*x^3 - x^4 + 2*x^5 + 2*x^6 + 2*x^7 - x^8 - 4*x^9 + ...
%t CoefficientList[Series[(1-x^2)^2*(1+x)^2/(1-x^6), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 12 2018 *)
%t a[ n_] := {2, -1, -4, -1, 2, 2}[[Mod[n, 6, 1]]] - Boole[n==0]; (* _Michael Somos_, Nov 14 2019 *)
%o (PARI) {a(n) = [2, 2, -1, -4, -1, 2][n%6 + 1] - (n == 0)};
%o (PARI) {a(n) = if( n<0, a(-n), polcoeff( (1 + x)^2 * (1 - x^2)^2 / (1 - x^6) + x * O(x^n), n))};
%o (PARI) {a(n) = if( n<0, a(-n), if( n<1, n==0, sumdiv( n, d, 2*(d==1) - 3*(d==2) - 6*(d==3) + 9*(d==6))))};
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2)^2*(1+x)^2/(1-x^6))); // _G. C. Greubel_, Aug 12 2018
%K sign
%O 0,2
%A _Michael Somos_, Apr 04 2012
|
