A259030 a(n) is multiplicative with a(2^e) = -(1 - (-1)^e) / 2 if e > 0, a(p^e) = Kronecker(5, p)^e if p > 2.

1, -1, -1, 0, 0, 1, -1, -1, 1, 0, 1, 0, -1, 1, 0, 0, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, -1

Offset: 1

Author

Michael Somos, Jun 17 2015

Keywords

• sign
• easy
• mult

Examples

			G.f. = x - x^2 - x^3 + x^6 - x^7 - x^8 + x^9 + x^11 - x^13 + x^14 - x^17 + ...
		

Crossrefs

Cf. A080891, A105368, A113185.

Programs

Mathematica

f[p_, e_] := KroneckerSymbol[5, p]^e; f[2, e_] := -(1 - (-1)^e) / 2; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

PARI

{a(n) = my(A, p, e); if( !n, 0, A = factor(abs(n)); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -(e%2), kronecker( 5, p)^e)))};

Formula

a(n) = a(-n) for all n in Z.

a(2*n + 1) = A105368(n). a(4*n + 1) = A080891(n-1). a(4*n + 2) = - A105368(n). a(4*n - 1) = A080891(n+1).

A113185(n) = Sum_{d|n} d * a(d) * -(-1)^(n/d) if n > 0.

G.f.: f(x) - Sum_{k>0} f(x^2^(2*k-1)) where f(x) := x * (1 - x^2) * (1 - x^6) / (1 - x^10).