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%I #41 Nov 17 2023 11:21:42
%S 1,1,1,1,2,1,0,1,1,2,0,1,2,0,2,1,2,1,0,2,0,0,0,1,3,2,1,0,2,2,0,1,0,2,
%T 0,1,2,0,2,2,2,0,0,0,2,0,0,1,1,3,2,2,2,1,0,0,0,2,0,2,2,0,0,1,4,0,0,2,
%U 0,0,0,1,2,2,3,0,0,2,0,2,1,2,0,0,4,0,2,0,2,2,0,0,0,0,0,1,2,1,0,3,2,2,0,2,0
%N a(n) = Sum_{d|n} Kronecker(-36, d).
%D Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 197, Entry 44.
%H G. C. Greubel, <a href="/A035154/b035154.txt">Table of n, a(n) for n = 1..1000</a>
%F Expansion of -1 + (theta_3(q)^2 + theta_3(q^3)^2) / 2 in powers of q. - _Michael Somos_, Jul 09 2013
%F From _Michael Somos_, Jul 30 2006: (Start)
%F Moebius transform is period 12 sequence [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, ...].
%F Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1(mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3(mod 4). (End)
%F Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -36, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -36, p) * p^-s)). - _Michael Somos_, Jun 24 2011
%F a(2*n) = a(3*n) = a(n). a(2*n + 1) = A125079(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).
%F 2 * a(n) = A122857(n) unless n=0. - _Michael Somos_, Jul 09 2013
%F G.f.: Sum_{n>=0} (-1)^n*( x^(6*n+1)/(1-x^(6*n+1)) + x^(6*n+5)/(1-x^(6*n+5)) ). - _Paul D. Hanna_, Dec 14 2011
%F G.f.: x/(1-x) + x^5/(1-x^5) - x^7/(1-x^7) - x^11/(1-x^11) + x^13/(1-x^13) + x^17/(1-x^17) --++ ...
%F a(n) = A002654(n) + A002654(3*n). - _Michael Somos_, Jan 25 2017
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - _Amiram Eldar_, Nov 17 2023
%e G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^12 + 2*x^13 + ...
%t a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Jun 24 2011 *)
%t a[ n_] := SeriesCoefficient[ (-2 + EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 4, {q, 0, n}]; (* _Michael Somos_, Jul 09 2013 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -36, d)))}; /* _Michael Somos_, Jul 30 2006 */
%o (PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -36, p) * X))) [n])}; /* _Michael Somos_, Jul 30 2006 */
%o (PARI) {a(n)=polcoeff(sum(m=0,n\6+1,(-1)^m*(x^(6*m+1)/(1-x^(6*m+1)+x*O(x^n)) + x^(6*m+5)/(1-x^(6*m+5)+x*O(x^n)))),n)} /* _Paul D. Hanna_ */
%Y Cf. A002654, A008441, A019670, A122856, A122857, A122865, A125079.
%K nonn,easy,mult
%O 1,5
%A _N. J. A. Sloane_
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