A227354 - OEIS

%I #24 Jun 12 2024 03:47:37

%S 1,12,-6,12,12,0,-6,24,-6,12,0,0,12,24,-12,0,12,0,-6,24,0,24,0,0,-6,

%T 12,-12,12,24,0,0,24,-6,0,0,0,12,24,-12,24,0,0,-12,24,0,0,0,0,12,36,

%U -6,0,24,0,-6,0,-12,24,0,0,0,24,-12,24,12,0,0,24,0,0,0

%N Expansion of 2 * a(q) - a(q^2) in powers of q where a() is a cubic AGM theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A227354/b227354.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F Expansion of (4 * b(q^4)^2 - 2 * b(q) * b(q^4) - b(q)^2) / b(q^2) in powers of q where b() is a cubic AGM theta function.

%F Expansion of phi(-q^2)^3 / phi(-q^6) + 12 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jan 09 2015

%F Expansion of theta_4(q^2)^3 / theta_4(q^6) + 3 * theta_2(q) * theta_2(q^3) in powers of q.

%F Moebius transform is period 6 sequence [ 12, -18, 0, 18, -12, 0, ...].

%F a(n) = 12 * b(n) where b(n) is multiplicative with b(2^e) = (1 + 3*(-1)^e) / 4, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

%F a(n) = A122859(8*n). a(2*n) = A122859(n). a(2*n + 1) = 12 * A033762(n). a(4*n) = a(n). a(4*n + 1) = 12 * A112604(n). a(4*n + 2) = -6 * A033762(n). a(4*n + 3) = 12 * A112605(n).

%F G.f.: 1 + 6 * Sum_{k>0} ((k mod 2) + 1) * x^k / (1 + x^k + x^(2*k)). - _Michael Somos_, Jan 09 2015

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - _Amiram Eldar_, Nov 23 2023

%e G.f. = 1 + 12*q - 6*q^2 + 12*q^3 + 12*q^4 - 6*q^6 + 24*q^7 - 6*q^8 + 12*q^9 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] + 3 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}];

%t a[ n_] := If[ n < 1, Boole[n == 0], 6 Sum[ JacobiSymbol[ d, 3] (Mod[ n/d, 2] + 1), {d, Divisors@n}]]; (* _Michael Somos_, Jan 09 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, 12 * sumdiv( n, d, kronecker( d, 3)) - 6 * sumdiv( 2*n, d, kronecker( d, 3)))};

%o (PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 12 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, (1 + 3*(-1)^e) / 4, if( p == 3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2 ))))))};

%Y Cf. A033762, A112604, A112605, A122859, A304656.

%Y Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

%K sign,easy

%O 0,2

%A _Michael Somos_, Jul 08 2013