A097723 One fourth of sum of divisors of 4n+3.

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

Offset: 0

Author

N. J. A. Sloane, Sep 11 2004

Keywords

• nonn

Comments

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

Examples

			1 + 2*x + 3*x^2 + 6*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 8*x^7 + 12*x^8 + ...
q^3 + 2*q^7 + 3*q^11 + 6*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 8*q^31 + ...
		

References

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

Links

• Ivan Neretin, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions.
• Min Wang, Zhi-Hong Sun, On the number of representations of n as a linear combination of four triangular numbers II, arXiv:1511.00478 [math.NT], 2015.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
• Kenneth S. Williams, n = delta+delta+2(delta+delta), Far East J. Math. Sci. 11 (2003), 233-240.

Crossrefs

Cf. A000203, A004767, A033686.

Programs

Mathematica

Table[DivisorSigma[1, 4n+3]/4, {n, 0, 72}] (* Jean-François Alcover, Nov 30 2015 *)

PARI

{a(n) = if( n<0, 0, sigma(4*n + 3) / 4)} /* Michael Somos, Jul 05 2006 */

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 */

Formula

Euler transform of period 4 sequence [2, 0, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004

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

Expansion of q^(-3/2) * (theta_2(q) * theta_2(q^2))^2 / 16 in powers of q^2. - Michael Somos, Jul 05 2006

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

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

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

a(n) = number of representations of n as the sum of two triangular numbers and twice two triangular numbers. - Michael Somos, Jul 05 2006

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

a(n) = A000203(A004767(n))/4. - Michel Marcus, Nov 30 2015

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