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A112610
Number of representations of n as a sum of two squares and two triangular numbers.
17
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
0,2
COMMENTS
Also row sums of A239931, hence the sequence has a symmetric representation. - Omar E. Pol, Aug 30 2015
LINKS
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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
EXAMPLE
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
MATHEMATICA
Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* Michael De Vlieger, Aug 31 2015 *)
PROG
(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 */
(Magma) [DivisorSigma(1, 4*n+1): n in [0..60]]; // Vincenzo Librandi, Sep 18 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
James A. Sellers, Dec 21 2005
STATUS
approved