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A244611 Expansion of (phi(q) + phi(q^2) - phi(q^3) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A244611 #16 Feb 16 2025 08:33:23
%S A244611 1,1,-1,1,0,-1,0,1,1,0,0,-1,0,0,0,1,0,1,0,0,0,0,0,-1,1,0,-1,0,0,0,0,1,
%T A244611 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,1,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,
%U A244611 0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,1
%N A244611 Expansion of (phi(q) + phi(q^2) - phi(q^3) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
%H A244611 Antti Karttunen, <a href="/A244611/b244611.txt">Table of n, a(n) for n = 1..65537</a>
%H A244611 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A244611 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A244611 Multiplicative with a(2^e) = 1, a(3^e) = (-1)^e, and a(p^e) = (1 + (-1)^e) / 2 if p>3.
%F A244611 G.f.: (theta_3(q) + theta_3(q^2) - theta_3(q^3) - theta_3(q^6)) / 2.
%F A244611 a(2*n) = a(n). a(2*n + 1) = A214505(n). a(3*n) = -a(n). a(3*n + 1) = A089801(n). a(6*n + 5) = 0.
%F A244611 From _Amiram Eldar_, Sep 12 2023: (Start)
%F A244611 Dirichlet g.f.: (1 + 1/2^s) * (1 - 1/3^s) * zeta(2*s).
%F A244611 Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1 + 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(6) = 0.721508... . (End)
%e A244611 G.f. = q + q^2 - q^3 + q^4 - q^6 + q^8 + q^9 - q^12 + q^16 + q^18 + ...
%o A244611 (PARI) {a(n) = issquare(n) + issquare(2*n) - issquare(3*n) - issquare(6*n)};
%o A244611 (PARI) {a(n) = if( n<1, 0, n/= 2^valuation(n, 2); issquare(n) - issquare(n*3))};
%o A244611 (PARI) {a(n) = local(A); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==2, 1, if( p==3, (-1)^e, !(e%2))))))};
%o A244611 (Scheme)
%o A244611 ;; Based on the given multiplicative formula, and using the memoization-macro definec:
%o A244611 (definec (A244611 n) (cond ((= 1 n) n) ((even? n) (A244611 (A000265 n))) ((zero? (modulo n 3)) (* (expt -1 (A067029 n)) (A244611 (A028234 n)))) (else (* 1/2 (+ 1 (expt -1 (A067029 n))) (A244611 (A028234 n))))))
%o A244611 ;; _Antti Karttunen_, Dec 07 2017
%Y A244611 Cf. A089801, A214505.
%K A244611 sign,easy,mult
%O A244611 1,1
%A A244611 _Michael Somos_, Jul 01 2014
%E A244611 More terms from _Antti Karttunen_, Dec 07 2017