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A185276 Kronecker symbol (-100 / n).

%I A185276 #32 Nov 14 2024 17:08:51
%S A185276 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,
%T A185276 -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,
%U A185276 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
%N A185276 Kronecker symbol (-100 / n).
%C A185276 This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and A289741 = {(-20/n)}. - _Jianing Song_, Nov 14 2024
%H A185276 Jianing Song, <a href="/A185276/b185276.txt">Table of n, a(n) for n = 0..1000</a>
%H A185276 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>, 2014.
%H A185276 <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 A185276 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 A185276 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 A185276 G.f.: x * (1 - x^2) * (1 - x^6) / (1 + x^10) = x / (1 + x^2) - x^5 / (1 + x^10).
%F A185276 a(n + 20) = -a(-n) = a(n). a(2*n) = a(5*n) = 0.
%F A185276 Dirichlet convolution with A000012 is A053694 offset 1.
%F A185276 Sum_{k=1..n} abs(a(k)) ~ 2*n/5. - _Amiram Eldar_, Jan 29 2024
%e A185276 x - x^3 - x^7 + x^9 - x^11 + x^13 + x^17 - x^19 + x^21 - x^23 - x^27 + ...
%t A185276 f[n_] := KroneckerSymbol[-100, n]; Array[f, 100] (* _Robert G. Wilson v_ *)
%o A185276 (PARI) {a(n) = kronecker( -100, n)}
%o A185276 (PARI) {a(n) = (n%2) * (-1) ^ (n\10) * kronecker( 5, n)}
%o A185276 (PARI) {a(n) = sign(n) * polcoeff( x * (1 - x^2) * (1 - x^6) / (1 + x^10) + x * O(x^abs(n)), abs(n))}
%o A185276 (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 A185276 Cf. A000012, A053694, A289741.
%K A185276 sign,easy,mult
%O A185276 0,1
%A A185276 _Michael Somos_, Feb 19 2011
%E A185276 a(0) prepended by _Jianing Song_, Nov 14 2024