A185276 Kronecker symbol (-100 / n).

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, 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, -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

Offset

1,1

Links

Table of n, a(n) for n=1..100.

Michael Somos, Rational Function Multiplicative Coefficients, 2014.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Formula

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).

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].

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

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

Dirichlet convolution with A000012 is A053694 offset 1.

Sum_{k=1..n} abs(a(k)) ~ 2*n/5. - Amiram Eldar, Jan 29 2024

Example

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

Mathematica

f[n_] := KroneckerSymbol[-100, n]; Array[f, 100] (* Robert G. Wilson v *)

Prog

(PARI) {a(n) = kronecker( -100, n)}
(PARI) {a(n) = (n%2) * (-1) ^ (n\10) * kronecker( 5, n)}
(PARI) {a(n) = sign(n) * polcoeff( x * (1 - x^2) * (1 - x^6) / (1 + x^10) + x * O(x^abs(n)), abs(n))}
(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 )))))}

Crossrefs

Cf. A000012, A053694.

Sequence in context: A015029 A014789 A014669 * A266282 A228747 A285162

Adjacent sequences: A185273 A185274 A185275 * A185277 A185278 A185279

Keyword

sign,mult

Author

Michael Somos, Feb 19 2011

Status

approved