A000603 Number of nonnegative solutions to x^2 + y^2 <= n^2.

1, 3, 6, 11, 17, 26, 35, 45, 58, 73, 90, 106, 123, 146, 168, 193, 216, 243, 271, 302, 335, 365, 402, 437, 473, 516, 557, 600, 642, 687, 736, 782, 835, 886, 941, 999, 1050, 1111, 1167, 1234, 1297, 1357, 1424, 1491, 1564, 1636, 1703, 1778, 1852, 1931, 2012, 2095

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Row sums of triangle A255238. - Wolfdieter Lang, Mar 15 2015

References

• H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
• 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

Crossrefs

Column k=2 of A302998.

Cf. A000328, A036695, A036702, A255238, A255195.

Programs

Haskell

a000603 n = length [(x,y) | x <- [0..n], y <- [0..n], x^2 + y^2 <= n^2]
-- Reinhard Zumkeller, Jan 23 2012

Mathematica

Table[cnt = 0; Do[If[x^2 + y^2 <= n^2, cnt++], {x, 0, n}, {y, 0, n}]; cnt, {n, 0, 51}] (* T. D. Noe, Apr 02 2013 *)
Table[If[n==1,1,2*Sum[Sum[A255195[[n, n - k + 1]], {k, 1, k}], {k, 1, n}] - Ceiling[(n - 1)/Sqrt[2]]],{n,1,52}] (* Mats Granvik, Feb 19 2015 *)

PARI

a(n)=my(n2=n^2);sum(a=0,n,sqrtint(n2-a^2)+1) \\ Charles R Greathouse IV, Apr 03 2013

Python

from math import isqrt
def A000603(n): return (m:=n<<1)+sum(isqrt(k*(m-k)) for k in range(1,n))+1 # Chai Wah Wu, Jul 18 2024

Formula

a(n) = n^2 * Pi/4 + O(n). - Charles R Greathouse IV, Apr 03 2013

a(n) = A001182(n) + 2*n + 1. - R. J. Mathar, Jan 07 2015

a(n) = 2*A026702(n) - (1 + floor(n/sqrt(2))), n >= 0. - Wolfdieter Lang, Mar 15 2015

a(n) = [x^(n^2)] (1 + theta_3(x))^2/(4*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

Extensions

More terms from David W. Wilson, May 22 2000