A097723 - OEIS

%I #31 Dec 17 2022 04:22:45

%S 1,2,3,6,5,6,10,8,12,14,11,12,18,18,15,26,17,18,31,20,21,30,28,30,39,

%T 26,27,38,36,36,42,32,33,60,35,42,57,38,48,54,41,42,65,62,45,62,54,48,

%U 84,50,60,78,53,66,74,56,57,96,72,60,91,70,63,108,76,66,90,68,93,104,71,84,98

%N One fourth of sum of divisors of 4n+3.

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

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 76, Eq. (31.54).

%H Ivan Neretin, <a href="/A097723/b097723.txt">Table of n, a(n) for n = 0..10000</a>

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

%H Min Wang, Zhi-Hong Sun, <a href="http://arxiv.org/abs/1511.00478">On the number of representations of n as a linear combination of four triangular numbers II</a>, arXiv:1511.00478 [math.NT], 2015.

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

%H Kenneth S. Williams, <a href="http://www.pphmj.com/abstract/1605.htm">n = delta+delta+2(delta+delta)</a>, Far East J. Math. Sci. 11 (2003), 233-240.

%F Euler transform of period 4 sequence [2, 0, 2, -4, ...]. - _Vladeta Jovovic_, Sep 14 2004

%F Expansion of q^(-3/4) * eta^2(q^2) * eta^4(q^4) / eta^2(q) in powers of q. - _Michael Somos_, Jul 05 2006

%F Expansion of q^(-3/2) * (theta_2(q) * theta_2(q^2))^2 / 16 in powers of q^2. - _Michael Somos_, Jul 05 2006

%F Expansion of (psi(x) * psi(x^2))^2 in powers of x where psi() is a Ramanujan theta function.

%F a(n) = sigma(4*n + 3) / 4 = A000203(4*n + 3) / 4.

%F a(n) = number of solutions of 8*n + 6 = x^2 + y^2 + 2*z^2 + 2*w^2 in positive odd integers.

%F a(n) = number of representations of n as the sum of two triangular numbers and twice two triangular numbers. - _Michael Somos_, Jul 05 2006

%F G.f.: (Product_{k>0} (1 - x^(4*k))^2 / (1 - x^(2*k - 1)))^2.

%F a(n) = A000203(A004767(n))/4. - _Michel Marcus_, Nov 30 2015

%F Sum_{k=0..n} a(k) = (Pi^2/16) * n^2 + O(n*log(n)). - _Amiram Eldar_, Dec 17 2022

%e 1 + 2*x + 3*x^2 + 6*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 8*x^7 + 12*x^8 + ...

%e q^3 + 2*q^7 + 3*q^11 + 6*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 8*q^31 + ...

%t Table[DivisorSigma[1, 4n+3]/4, {n, 0, 72}] (* _Jean-François Alcover_, Nov 30 2015 *)

%o (PARI) {a(n) = if( n<0, 0, sigma(4*n + 3) / 4)} /* _Michael Somos_, Jul 05 2006 */

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A))^2, n))} /* _Michael Somos_, Jul 05 2006 */

%Y Cf. A000203, A004767, A033686.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Sep 11 2004