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%I M3829 N1570 #76 Aug 04 2022 05:19:46
%S 1,5,13,29,49,81,113,149,197,253,317,377,441,529,613,709,797,901,1009,
%T 1129,1257,1373,1517,1653,1793,1961,2121,2289,2453,2629,2821,3001,
%U 3209,3409,3625,3853,4053,4293,4513,4777,5025,5261,5525,5789,6077,6361,6625
%N Number of points of norm <= n^2 in square lattice.
%C Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
%D H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
%D C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 47.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe and Robert Israel, <a href="/A000328/b000328.txt">Table of n, a(n) for n = 0..10000</a> (n=0..1000 from T. D. Noe)
%H W. Fraser and C. C. Gotlieb, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0155788-9">A calculation of the number of lattice points in the circle and sphere</a>, Math. Comp., 16 (1962), 282-290.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem</a>
%F a(n) = 1 + 4 * Sum_{j>=0} floor(n^2/(4*j+1)) - floor(n^2/(4*j+3)). Also a(n) = A057655(n^2). - _Max Alekseyev_, Nov 18 2007
%F a(n) = 4*A000603(n) - (4*n+3), n >= 0. - _Wolfdieter Lang_, Mar 15 2015
%F a(n) = 1+4*n^2-4*ceiling((n-1)/sqrt(2))-8*A247588(n-1), n>1. - _Mats Granvik_, May 23 2015
%F a(n) = [x^(n^2)] theta_3(x)^2/(1 - x), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Apr 14 2018
%t Table[Sum[SquaresR[2, k], {k, 0, n^2}], {n, 0, 46}]
%o (PARI) { a(n) = 1 + 4 * sum(j=0,n^2\4, n^2\(4*j+1) - n^2\(4*j+3) ) } /* _Max Alekseyev_, Nov 18 2007 */
%o (Haskell)
%o a000328 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n^2]
%o -- _Reinhard Zumkeller_, Jan 23 2012
%o (Python)
%o def A000328(n):
%o return (sum([int((n**2 - y**2)**0.5) for y in range(1, n)]) * 4 + 4*n + 1)
%o # _Karl-Heinz Hofmann_, Aug 03 2022
%Y Column k=2 of A302997.
%Y Equals A051132 + A046109. For another version see A057655.
%Y Cf. A093832, A093836, A093837, A000603, A255238, A305575, A305576.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_, May 22 2000
%E Edited at the suggestion of _Max Alekseyev_ by _N. J. A. Sloane_, Nov 18 2007
%E Incorrect comment removed by _Eric M. Schmidt_, May 28 2015
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