

A000328


Number of points of norm <= n^2 in square lattice.
(Formerly M3829 N1570)


20



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
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OFFSET

0,2


COMMENTS

Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.
Or, numerator of N(r)/r^2, where N(r) is the number of lattice points inside a circle of radius r.


REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, p. 106.
W. Fraser and C. C. Gotlieb, A calculation of the number of lattice points in the circle and sphere, Math. Comp., 16 (1962), 282290.
H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 3563.
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, Table of n, a(n) for n=0..1000
Eric Weisstein's World of Mathematics, Gauss's Circle Problem


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


MATHEMATICA

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


PROG

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


CROSSREFS

Equals A051132 + A046109. For another version see A057655.
Cf. A093832, A093837.
Sequence in context: A095085 A230281 A093836 * A100438 A129371 A212008
Adjacent sequences: A000325 A000326 A000327 * A000329 A000330 A000331


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

More terms from David W. Wilson, May 22, 2000
Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev.


STATUS

approved



