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A055410 Number of points in Z^4 of norm <= n.

%I A055410 #35 Jun 24 2024 10:07:33
%S A055410 1,9,89,425,1281,3121,6577,11833,20185,32633,49689,72465,102353,
%T A055410 140945,190121,250553,323721,411913,519025,643441,789905,961721,
%U A055410 1156217,1380729,1638241,1927297,2257281,2624417,3035033,3490601,4000425
%N A055410 Number of points in Z^4 of norm <= n.
%H A055410 Chai Wah Wu, <a href="/A055410/b055410.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..500 from Andrew Howroyd)
%F A055410 a(n) = A046895(n^2). - _Joerg Arndt_, Apr 08 2013
%F A055410 a(n) = [x^(n^2)] theta_3(x)^4/(1 - x), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Apr 14 2018
%t A055410 a[n_] := SeriesCoefficient[EllipticTheta[3, 0, x]^4/(1 - x), {x, 0, n^2}];
%t A055410 a /@ Range[0, 30] (* _Jean-François Alcover_, Sep 23 2019, after _Ilya Gutkovskiy_ *)
%o A055410 (C)
%o A055410 int A055410(int i)
%o A055410 {
%o A055410     const int ring = i*i;
%o A055410     int result = 0;
%o A055410     for(int a = -i; a <= i; a++)
%o A055410         for(int b = -i; b <= i; b++)
%o A055410             for(int c = -i; c <= i; c++)
%o A055410                 for(int d = -i; d <= i; d++)
%o A055410                     if ( ring >= a*a + b*b + c*c + d*d ) result++;
%o A055410     return result;
%o A055410 } /* _Oskar Wieland_, Apr 08 2013 */
%o A055410 (PARI)
%o A055410 N=66;  q='q+O('q^(N^2));
%o A055410 t=Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q)); /* A046895 */
%o A055410 vector(sqrtint(#t),n,t[(n-1)^2+1])
%o A055410 /* _Joerg Arndt_, Apr 08 2013 */
%o A055410 (Python)
%o A055410 from math import isqrt
%o A055410 def A055410(n): return 1+((-(s:=n**2)*(n+1)+sum((q:=s//k)*((k<<1)+q+1) for k in range(1,n+1))&-1)<<2)+(((t:=isqrt(m:=s>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # _Chai Wah Wu_, Jun 24 2024
%Y A055410 Column k=4 of A302997.
%Y A055410 Cf. A046895 (sizes of successive clusters in Z^4 lattice).
%K A055410 nonn
%O A055410 0,2
%A A055410 _David W. Wilson_