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A055412 Number of points in Z^6 of norm <= n.

%I A055412 #27 Jun 21 2024 12:40:11
%S A055412 1,13,485,4197,23793,84769,252673,622573,1395261,2787125,5260181,
%T A055412 9249417,15637897,25112577,39258381,59174749,87380293,125264525,
%U A055412 176663297,244000537,332379769,444344469,587923621,766764301,990981473
%N A055412 Number of points in Z^6 of norm <= n.
%H A055412 Andrew Howroyd, <a href="/A055412/b055412.txt">Table of n, a(n) for n = 0..500</a>
%F A055412 a(n) = A122510(6,n^2). - _R. J. Mathar_, Apr 21 2010
%F A055412 a(n) = [x^(n^2)] theta_3(x)^6/(1 - x), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Apr 14 2018
%t A055412 t[d_, n_] := t[d, n] = t[d, n - 1] + SquaresR[d, n]; t[d_, 0] = 1;
%t A055412 a[n_] := t[6, n^2];
%t A055412 a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 27 2019, after _R. J. Mathar_ *)
%o A055412 (Python)
%o A055412 from math import prod
%o A055412 from sympy import factorint
%o A055412 def A055412(n):
%o A055412     c = 1
%o A055412     for m in range(1,n**2+1):
%o A055412         f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(m).items()]
%o A055412         c += (prod((p**(e+1<<1)-a)//(p**2-a) for p, e, a in f)<<2)-prod(((k:=p**2*a)**(e+1)-1)//(k-1) for p, e, a in f)<<2
%o A055412     return c # _Chai Wah Wu_, Jun 21 2024
%Y A055412 Column k=6 of A302997.
%Y A055412 Cf. A122510.
%K A055412 nonn
%O A055412 0,2
%A A055412 _David W. Wilson_