A112610 Number of representations of n as a sum of two squares and two triangular numbers.

1, 6, 13, 14, 18, 32, 31, 30, 48, 38, 42, 78, 57, 54, 80, 62, 84, 96, 74, 96, 121, 108, 90, 128, 98, 102, 192, 110, 114, 182, 133, 156, 176, 160, 138, 192, 180, 150, 234, 158, 192, 288, 183, 174, 240, 182, 228, 320, 194, 198, 272, 252, 240, 288, 256, 252, 403, 230

Offset: 0

Author

James Sellers, Dec 21 2005

Keywords

• nonn

Comments

Also row sums of A239931, hence the sequence has a symmetric representation. - Omar E. Pol, Aug 30 2015

Examples

			a(1) = 6 since we can write 1 = 1^2 + 0^2 + 0 + 0 = (-1)^2 + 0^2 + 0 + 0 = 0^2 + 1^2 + 0 + 0 = 0^2 + (-1)^2 + 0 + 0 = 0^2 + 0^2 + 1 + 0 = 0^2 + 0^2 + 0 + 1
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10001
• M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Crossrefs

Cf. A000203, A193553, A237593, A239052, A239053, A239931.

Programs

Magma

[DivisorSigma(1, 4*n+1): n in [0..60]]; // Vincenzo Librandi, Sep 18 2015

Mathematica

Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* Michael De Vlieger, Aug 31 2015 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^14/eta(x+A)^6/eta(x^4+A)^4, n))} /* Michael Somos, Jul 04 2006 */

Formula

a(n) = sigma(4n+1) where sigma(n) = A000203(n) is the sum of the divisors of n.

Euler transform of period 4 sequence [ 6, -8, 6, -4, ...]. - Michael Somos, Jul 04 2006

Expansion of q^(-1/4)eta^14(q^2)/(eta^6(q)eta^4(q^4)) in powers of q. - Michael Somos, Jul 04 2006

Expansion of psi(q)^2*phi(q)^2, i.e., convolution of A004018 and A008441 [Hirschhorn]. - R. J. Mathar, Mar 24 2011

Sum_{k=0..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022