%I #8 Mar 12 2021 22:24:49

%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,

%T -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,

%U 1,-1,2,-1,1,-1,0,-1,1,-1,2,-1,1,-1,0,-1,1,-1,2,-1,1

%N Expansion of (1 - x^5) / ((1 + x) * (1 + x^4)) in powers of x.

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,0,-1,-1).

%F a(n) = -b(n) and b() is multiplicative with b(2) = -1, b(4) = 0, b(2^e) = -2 if e>2, b(p^e) = 1 if p>2.

%F Euler transform of length 8 sequence [-1, 1, 0, -1, -1, 0, 0, 1].

%F Moebius transform is length 8 sequence [-1, 2, 0, -1, 0, 0, 0, 2].

%F G.f.: (1 - x^5) / ((1 + x) * (1+ x^4)) = -1 + 1 / (1 + x) + 1 / (1 + x^4).

%F a(n) = a(-n) for all n in Z. a(n+4) = a(n-4) except if n=4 or n=-4.

%F a(n) = (-1)^n * A257179(n), a(2*n + 1) = -1, a(4*n + 2) = 1, a(8*n + 4) = 0 for all n in Z.

%e G.f. = 1 - x + x^2 - x^3 - x^5 + x^6 - x^7 + 2*x^8 - x^9 + x^10 + ...

%t a[ n_] := (-1)^n + If[Mod[n, 4] == 0, (-1)^(n/4), 0] - Boole[n == 0];

%t a[ n_] := {-1, 1, -1, 0, -1, 1, -1, 2}[[Mod[n, 8, 1]]] - Boole[n == 0];

%t a[ n_] := SeriesCoefficient[ (1 - x^5) / ((1 + x) * (1 + x^4)), {x, 0, Abs@n}];

%o (PARI) {a(n) = (-1)^n + if(n%4==0, (-1)^(n/4)) - (n==0)};

%o (PARI) {a(n) = [2, -1, 1, -1, 0, -1, 1, -1][n%8 + 1] - (n==0)};

%o (PARI) {a(n) = n=abs(n); polcoeff( (1 - x^5) / ((1 + x) * (1 + x^4)) + x * O(x^n), n)};

%o (PARI) {a(n) = my(e); n=abs(n); if( n<1, n==0, e=valuation(n, 2); -if( e<3, [1, -1, 0][e+1], -2))};

%Y Cf. A257179.

%K sign,easy

%O 0,9

%A _Michael Somos_, Jan 06 2019