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Expansion of (1 - x) * (1 - x^3) / (1 - x^4) in powers of x.
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%I #13 Nov 01 2022 04:33:38

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

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

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

%N Expansion of (1 - x) * (1 - x^3) / (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_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1).

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

%F Euler transform of length 4 sequence [-1, 0, -1, 1].

%F Moebius transform is length 4 sequence [-1, 1, 0, 2].

%F G.f.: (1 - x) * (1 - x^3) / (1 - x^4) = -1 + 1 / (1 + x) + 1 / (1 + x^2).

%F a(n) = a(-n) for all n in Z. a(n+2) = a(n-2) except if n=2 or n=-2.

%F a(n) = (-1)^n * A098178(n), a(2*n + 1) = -1, a(4*n + 2) = 0 for all n in Z.

%e G.f. = 1 - x - x^3 + 2*x^4 - x^5 - x^7 + 2*x^8 - x^9 - x^11 + ...

%t a[ n_] := (-1)^n + If[Mod[n, 2] == 0, (-1)^(n/2), 0] - Boole[n == 0];

%t a[ n_] := {-1, 0, -1, 2}[[Mod[n, 4, 1]]] - Boole[n == 0];

%t a[ n_] := SeriesCoefficient[ (1 - x) (1 - x^3) / (1 - x^4), {x, 0, Abs@n}];

%t LinearRecurrence[{-1,-1,-1},{1,-1,0,-1},80] (* _Harvey P. Dale_, May 31 2021 *)

%o (PARI) {a(n) = (-1)^n + if(n%2==0, (-1)^(n/2)) - (n==0)};

%o (PARI) {a(n) = [2, -1, 0, -1][n%4 + 1] - (n==0)};

%o (PARI) {a(n) = n = abs(n); polcoeff( (1 - x) * (1 - x^3) / (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<2, 1-e, -2))};

%Y Cf. A098178

%K sign,easy

%O 0,5

%A _Michael Somos_, Jan 06 2019