The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A000328 Number of points of norm <= n^2 in square lattice. (Formerly M3829 N1570) 40
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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 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 (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 CROSSREFS Column k=2 of A302997. Equals A051132 + A046109. For another version see A057655. Cf. A093832, A093836, A093837, A000603, A255238, A305575, A305576. Sequence in context: A309371 A230281 A093836 * A272750 A272801 A100438 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 at the suggestion of Max Alekseyev by N. J. A. Sloane, Nov 18 2007 Incorrect comment removed by Eric M. Schmidt, May 28 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 25 07:08 EDT 2024. Contains 372782 sequences. (Running on oeis4.)