A112610 - OEIS

%I #41 Dec 17 2022 04:22:49

%S 1,6,13,14,18,32,31,30,48,38,42,78,57,54,80,62,84,96,74,96,121,108,90,

%T 128,98,102,192,110,114,182,133,156,176,160,138,192,180,150,234,158,

%U 192,288,183,174,240,182,228,320,194,198,272,252,240,288,256,252,403,230

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

%C Also row sums of A239931, hence the sequence has a symmetric representation. - _Omar E. Pol_, Aug 30 2015

%H Amiram Eldar, <a href="/A112610/b112610.txt">Table of n, a(n) for n = 0..10001</a>

%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.

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

%F Euler transform of period 4 sequence [ 6, -8, 6, -4, ...]. - _Michael Somos_, Jul 04 2006

%F 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

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

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

%e 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

%t Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* _Michael De Vlieger_, Aug 31 2015 *)

%o (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 */

%o (Magma) [DivisorSigma(1, 4*n+1): n in [0..60]]; // _Vincenzo Librandi_, Sep 18 2015

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

%K nonn

%O 0,2

%A _James A. Sellers_, Dec 21 2005