A000328 Number of points of norm <= n^2 in square lattice.

1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441, 529, 613, 709, 797, 901, 1009, 1129, 1257, 1373, 1517, 1653, 1793, 1961, 2121, 2289, 2453, 2629, 2821, 3001, 3209, 3409, 3625, 3853, 4053, 4293, 4513, 4777, 5025, 5261, 5525, 5789, 6077, 6361, 6625

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
• H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
• C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 47.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe and Robert Israel, Table of n, a(n) for n = 0..10000 (n=0..1000 from T. D. Noe)
• W. Fraser and C. C. Gotlieb, A calculation of the number of lattice points in the circle and sphere, Math. Comp., 16 (1962), 282-290.
• Eric Weisstein's World of Mathematics, Gauss's Circle Problem
• Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. See Table 3 p. 18.

Crossrefs

Column k=2 of A302997.

Equals A051132 + A046109. For another version see A057655.

Cf. A093832, A093836, A093837, A000603, A255238, A305575, A305576.

Programs

Haskell

a000328 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n^2]
-- Reinhard Zumkeller, Jan 23 2012

Mathematica

Table[Sum[SquaresR[2, k], {k, 0, n^2}], {n, 0, 46}]

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 */

Python

def A000328(n):
    return (sum([int((n**2 - y**2)**0.5) for y in range(1, n)]) * 4 + 4*n + 1)
    # Karl-Heinz Hofmann, Aug 03 2022

Python

from math import isqrt
def A000328(n): return 1+(sum(isqrt(k*((n<<1)-k)) for k in range(1,n+1))<<2) # Chai Wah Wu, Feb 12 2025

Formula

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

a(n) = 4*A000603(n) - (4*n+3), n >= 0. - Wolfdieter Lang, Mar 15 2015

a(n) = 1+4*n^2-4*ceiling((n-1)/sqrt(2))-8*A247588(n-1), n>1. - Mats Granvik, May 23 2015

a(n) = [x^(n^2)] theta_3(x)^2/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018

Limit_{n->oo} a(n)/n^2 = Pi. - Chai Wah Wu, Feb 12 2025

Extensions

More terms from David W. Wilson, May 22 2000

Edited at the suggestion of Max Alekseyev by N. J. A. Sloane, Nov 18 2007

Incorrect comment removed by Eric M. Schmidt, May 28 2015