A255238 -id:A255238 - cp's OEIS frontend

A255239 Alternating row sums of triangle A255238.

1, 1, 2, 3, 3, 4, 3, 5, 4, 7, 6, 8, 7, 10, 6, 9, 6, 11, 9, 14, 11, 15, 10, 15, 11, 14, 11, 18, 14, 19, 14, 18, 13, 20, 15, 23, 16, 25, 21, 24, 19, 23, 16, 27, 20, 28, 21, 30, 22, 27, 22, 31, 25, 34, 27, 35, 24, 35, 26, 32, 29, 36, 26, 39, 28

Offset: 0

Wolfdieter Lang, Mar 12 2015

See A255238 for a Gauss' circle problem comment and a link.

Cf. A255238.

a(n) = sum((-1)^m*T(n, m), m = 0..n), n >= 0, with T(n, m) = A255238(n, m) = 1 + floor(sqrt(n^2 - m^2)).

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

Column k=2 of A302997.
Equals A051132 + A046109. For another version see A057655.
Cf. A093832, A093836, A093837, A000603, A255238, A305575, A305576.

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

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

{ 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 */

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

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

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

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

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

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

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).

T. D. Noe, Table of n, a(n) for n = 0..1000

Column k=2 of A302998.

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

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

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 *)

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

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

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

More terms from David W. Wilson, May 22 2000

A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 6, 4, 3, 2, 7, 5, 4, 3, 1, 8, 6, 5, 4, 2, 9, 7, 6, 5, 3, 2, 10, 8, 7, 6, 5, 3, 1, 11, 9, 8, 7, 6, 4, 3, 1, 12, 10, 9, 8, 7, 5, 4, 2, 13, 11, 10, 9, 8, 6, 5, 3, 1, 14, 12, 11, 10, 9, 8, 6, 4, 3, 1, 15, 13, 12, 11, 10, 9, 7, 6, 4, 2, 16, 14, 13, 12, 11, 10, 8, 7, 5, 4, 2

Offset: 0

Wolfdieter Lang, Mar 14 2015

The row length of this array (irregular triangle) is 1 + flpoor(n/sqrt(2)) = 1 + A049472(n) = 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, ...
This entry is motivated by the proposal A255195 by Mats Granvik, who gave the first differences of this array.
See the MathWorld link on Gauss's circle problem.
The first octant of a square lattice (x, y) with n = x >= y = m >= 0 is considered. The number of lattice points in this octant covered by a circular disk of radius R = n around the origin having ordinate value y = m are denoted by T(n, m), for n >= 0 and m = 0, 1, ..., floor(n/sqrt(2)).
The row sums give RS(n) = A036702(n), n >= 0. This is the total number of square lattice points in the first octant covered by a circular disk of radius R = n.
The alternating row sums give A256094(n), n >= 0.
The total number of square lattice points in the first quadrant covered by a circular disk of radius R = n is therefore 2*RS(n) - (1 + floor(n/sqrt(2))) = A000603(n).

			The array (irregular triangle) T(n, m) begins:
n\m  0  1  2  3  4  5 6 7 8 9 10 ....
0:   1
1:   2
2:   3  1
3:   4  2  1
4:   5  3  2
5:   6  4  3  2
6:   7  5  4  3  1
7:   8  6  5  4  2
8:   9  7  6  5  3  2
9:  10  8  7  6  5  3 1
10: 11  9  8  7  6  4 3 1
11: 12 10  9  8  7  5 4 2
12: 13 11 10  9  8  6 5 3 1
13: 14 12 11 10  9  8 6 4 3 1
14: 15 13 12 11 10  9 7 6 4 2
15: 16 14 13 12 11 10 8 7 5 4  2
...

E. W. Weisstein, World of Mathematics, Gauss's Circle Problem.

Cf. A036702, A256094, A000328, A049472, A255195, A255238 (quadrant).

T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)).

cp's OEIS Frontend

A255239 Alternating row sums of triangle A255238.

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

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

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Haskell

Mathematica

PARI

Python

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A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

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