A000603 Number of nonnegative solutions to x^2 + y^2 <= n^2. (Formerly M2541 N1004)

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

0,2

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

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

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 *)

Prog

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

Crossrefs

Column k=2 of A302998.

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

Sequence in context: A247586 A377407 A107957 * A003453 A011901 A169739

Adjacent sequences: A000600 A000601 A000602 * A000604 A000605 A000606

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, May 22 2000

Status

approved