%I #25 Jan 29 2024 01:55:41

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

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

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

%N Kronecker symbol (-100 / n).

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>, 2014.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,-1).

%F a(n) is multiplicative with a(2^e) = a(5^e) = 0^e, a(p^e) = 1 if p == 1 (mod 4) and p>5, a(p^e) = (-1)^e if p == 3 (mod 4).

%F Euler transform of length 20 sequence [ 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1].

%F G.f.: x * (1 - x^2) * (1 - x^6) / (1 + x^10) = x / (1 + x^2) - x^5 / (1 + x^10).

%F a(n + 20) = -a(-n) = a(n). a(2*n) = a(5*n) = 0.

%F Dirichlet convolution with A000012 is A053694 offset 1.

%F Sum_{k=1..n} abs(a(k)) ~ 2*n/5. - _Amiram Eldar_, Jan 29 2024

%e x - x^3 - x^7 + x^9 - x^11 + x^13 + x^17 - x^19 + x^21 - x^23 - x^27 + ...

%t f[n_] := KroneckerSymbol[-100, n]; Array[f, 100] (* _Robert G. Wilson v_ *)

%o (PARI) {a(n) = kronecker( -100, n)}

%o (PARI) {a(n) = (n%2) * (-1) ^ (n\10) * kronecker( 5, n)}

%o (PARI) {a(n) = sign(n) * polcoeff( x * (1 - x^2) * (1 - x^6) / (1 + x^10) + x * O(x^abs(n)), abs(n))}

%o (PARI) {a(n) = local( A, p, e); if( n==0, 0, A = factor( abs(n)); sign(n) * prod( k=1, matsize( A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==2 | p==5, 0, if( p%4==1, 1, (-1)^e )))))}

%Y Cf. A000012, A053694.

%K sign,mult

%O 1,1

%A _Michael Somos_, Feb 19 2011