A000603 -id:A000603 - cp's OEIS frontend

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

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

A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2.

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  1,  2,   3,   4,    5,     6,  ...
  1,  3,   6,  11,   20,    36,  ...
  1,  4,  11,  29,   70,   157,  ...
  1,  5,  17,  54,  165,   482,  ...
  1,  6,  26,  99,  357,  1203,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.
Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425.
Main diagonal gives A302863.
Cf. A000122, A122510, A302996, A302997.

Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.

A048149 Array T read by diagonals: T(i,j) = number of pairs (h,k) with h^2+k^2 <= i^2+j^2, h>=0, k >= 0.

Original entry on oeis.org

1, 3, 3, 6, 4, 6, 11, 8, 8, 11, 17, 13, 9, 13, 17, 26, 19, 15, 15, 19, 26, 35, 28, 22, 20, 22, 28, 35, 45, 37, 30, 26, 26, 30, 37, 45, 58, 48, 39, 33, 31, 33, 39, 48, 58, 73, 62, 52, 43, 41, 41, 43, 52, 62, 73, 90, 75, 64, 54, 50, 48, 50, 54, 64, 75, 90

Offset: 0

Clark Kimberling

			Seen as a triangle:
[0]  1;
[1]  3,  3;
[2]  6,  4,  6;
[3] 11,  8,  8, 11;
[4] 17, 13,  9, 13, 17;
[5] 26, 19, 15, 15, 19, 26;
[6] 35, 28, 22, 20, 22, 28, 35;
[7] 45, 37, 30, 26, 26, 30, 37, 45;
[8] 58, 48, 39, 33, 31, 33, 39, 48, 58;
[9] 73, 62, 52, 43, 41, 41, 43, 52, 62, 73;

Cf. A000603 (right diagonal).

A048149 := proc(n, k) option remember; ## n = 0 .. infinity and k = 0 .. n
    local x, y, radius, nTotal;
    if n >= k then
        radius := floor(sqrt(n^2 + k^2));
        nTotal := 0;
        for x from 0 to radius do
            nTotal := nTotal + floor(sqrt(n^2 + k^2 - x^2)) + 1;
        end do;
        return nTotal;
    else
        return A048149(k, n);
    end if;
end proc: # Yu-Sheng Chang, Jan 14 2020

t[i_, j_] := Module[{h, k}, Reduce[h^2 + k^2 <= i^2 + j^2 && h >= 0 && k >= 0, {h, k}, Integers] // ToRules // Length[{##}]&]; Table[t[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)

a(55) corrected by Jean-François Alcover, Nov 26 2013

a(55) restored by Yu-Sheng Chang, Jan 14 2020

A036702 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b<=a.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 20, 25, 32, 40, 49, 57, 66, 78, 89, 102, 114, 128, 142, 158, 175, 190, 209, 227, 245, 267, 288, 310, 331, 354, 379, 402, 429, 455, 483, 512, 538, 569, 597, 631, 663, 693, 727, 761, 798, 834, 868, 906, 943, 983

Offset: 0

Clark Kimberling

Row sums of the irregular triangle A255250. - Wolfdieter Lang, Mar 15 2015

Index entries for Gaussian integers and primes

Cf. A036700, A049472, A000603, A000328, A255250.

A036702 := proc(n)
        local a,x,y ;
        a := 0 ;
        for x from 0 do
                if x^2 > n^2 then
                        return a;
                fi ;
                for y from 0 to x do
                        if y^2+x^2 <= n^2 then
                                a := a+1 ;
                        end if;
                end do;
        end do:
end proc: # R. J. Mathar, Oct 29 2011

a[n_] := Module[{a, b}, If[n == 0, 1, Reduce[a^2 + b^2 <= n^2 && a >= 0 && 0 <= b <= a, {a, b}, Integers] // Length]];
a /@ Range[0, 49] (* Jean-François Alcover, Oct 17 2019 *)

a(n) - A036700(n) = 1+A049472(n). - R. J. Mathar, Oct 29 2011

a(n) = sum(floor(sqrt(n^2 - m^2)) - (m-1), m = 0.. floor(n/sqrt(2))), n >= 0. See A255250. - Wolfdieter Lang, Mar 15 2015

A228233 Number of Gaussian primes of norm less than or equal to n in the first quadrant.

Original entry on oeis.org

0, 1, 5, 7, 9, 11, 17, 21, 23, 27, 35, 37, 41, 47, 49, 55, 63, 69, 77, 83, 91, 97, 103, 109, 119, 127, 133, 143, 151, 159, 169, 179, 187, 199, 209, 219, 227, 237, 245, 251, 265, 279, 287, 301, 311, 323, 335, 351, 367, 377, 385, 401, 419, 431, 441, 455, 469

Offset: 1

Olivier Gérard, Aug 17 2013

nonn

Include 2 times the primes (once for the real axis, once for the imaginary axis).

More precisely, a(n) includes all Gaussian primes (with the appropriate norms) on the first quadrant's bounding semi-axes. All such Gaussian primes occur in pairs {p, pi} (one real and one imaginary associate), where p is a classical prime of the form 4m + 3 (so p is in A002145) and p <= n. - Rick L. Shepherd, Jun 16 2017

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000603 (number of Gaussian integers in the first quadrant with norm less than or equal to n).
Cf. A062711 (counts the Gaussian primes on only one axis).
Cf. A228232 (this sequence excluding classical primes and pure imaginary primes).
Cf. A002145 (Gaussian primes that are positive integers).

nn = 100; t = Select[Flatten[Table[a + b*I, {a, 0, nn}, {b, 0, nn}]], PrimeQ[#, GaussianIntegers -> True] &]; t2 = Table[0, {nn}]; Do[f = Ceiling[Abs[i]]; If[f <= nn, t2[[f]]++], {i, t}]; Accumulate[t2] (* T. D. Noe, Aug 19 2013 *)

A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 2, 1, 1, 0, 2, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 2, 0, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Feb 16 2015

The sum of terms of row n is n.

Total of partial sums in reverse (from right to left) equals one eighth of the Gauss circle problem. Whenever there is the number 2 the border of the circle makes a jump upwards. Predicting where the 2's are would say something about the Gauss circle problem. The number of 2's equals the number of 0's in the same row, and is counted by A194920(n-1).

			1,
2, 0,
2, 1, 0,
2, 1, 1, 0,
2, 1, 2, 0, 0,
2, 1, 1, 2, 0, 0,
2, 1, 1, 2, 1, 0, 0,
2, 1, 1, 2, 2, 0, 0, 0,
2, 1, 1, 2, 1, 2, 0, 0, 0,
2, 1, 1, 1, 2, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0

E. W. Weisstein, Gauss's Circle Problem

Cf. A000603, A194920.

Flatten[Table[Sum[Table[If[And[If[n^2 + k^2 <= r^2, If[n >= k, 1, 0], 0] == 1, If[(n + 1)^2 + (k + 1)^2 <= r^2, If[n >= k, 1, 0], 0]== 0], 1, 0], {k, 0, r}], {n, 0, r}], {r, 0, 12}]]

A000603(n) = 2*(Sum_{k=1..n} Sum_{k=1..k} T(n,n-k+1))-ceiling((n-1)/sqrt(2)) for n>1.

A247588(n-1) = (Sum_{k=1..n} Sum_{k=1..k} (T(n,k) - T(n,n-k+1))/2).

A255238 Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m in the first quadrant.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 4, 1, 8, 7, 7, 7, 6, 5, 4, 1, 9, 8, 8, 8, 7, 7, 6, 4, 1, 10, 9, 9, 9, 9, 8, 7, 6, 5, 1, 11, 10, 10, 10, 10, 9, 9, 8, 7, 5, 1

Offset: 0

Wolfdieter Lang, Mar 12 2015

This entry is motivated by the proposal A255195 by Mats Granvik.
See the MathWorld link on Gauss's circle problem.
The first quadrant of a square lattice (x, y) with x = n >= 0, y = m >= 0, is considered. The number of lattice points 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, ..., n.
The same numbers occur if x and y are interchanged.
One could also consider the row reversed triangle.
The row sums give R(n) = A000603(n), n >= 0.
The alternating row sums give A255239(n), n >= 0.
The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3).

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

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

Cf. A000603, A000328, A255239.

T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n.

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 6, 29, 165, 1203, 9763, 83877, 793049, 7903501, 83570177, 933697153, 10905583809, 133352809334, 1695473999478, 22354920990148, 305096197935075, 4296142551821184, 62336908825014452, 930284705538262688, 14255992611680074754, 224065160215526683317, 3607018540134004189466

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n^2.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A302998.

Cf. A000603, A000604, A010052, A055403, A055404, A055405, A055406, A055407, A055408, A055409, A298329, A302861, A302862.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 22}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n^2}], {n, 0, 22}]

A036695 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n, b>=0.

Original entry on oeis.org

1, 4, 9, 18, 29, 46, 63, 82, 107, 136, 169, 200, 233, 278, 321, 370, 415, 468, 523, 584, 649, 708, 781, 850, 921, 1006, 1087, 1172, 1255, 1344, 1441, 1532, 1637, 1738, 1847, 1962, 2063, 2184, 2295, 2428, 2553, 2672, 2805, 2938

Offset: 0

Clark Kimberling

nonn

Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2 and y >= 0. [Reinhard Zumkeller, Jan 23 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for Gaussian integers and primes

Cf. A000603, A000328, A036696.

a036695 n = length [(x,y) | x <- [-n..n], y <- [0..n], x^2 + y^2 <= n^2]
-- Reinhard Zumkeller, Jan 23 2012

a[n_] := (k = 0; Do[If[x^2 + y^2 <= n^2, k++], {x, -n, n}, {y, 0, n}]; k); Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 08 2016 *)

Partial sums of A036696. - Sean A. Irvine, Nov 22 2020

A349609 Number of solutions to x^2 + y^2 <= n^2, where x, y are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 8, 8, 13, 15, 20, 22, 28, 31, 39, 43, 52, 54, 64, 69, 79, 83, 96, 102, 112, 121, 135, 140, 154, 162, 179, 185, 203, 212, 228, 238, 255, 265, 281, 296, 316, 326, 349, 359, 382, 394, 416, 429, 451, 469, 494, 508, 532, 547, 573, 587

Offset: 0

Ilya Gutkovskiy, Nov 23 2021

nonn

			a(4) = 3 since there are solutions (1,1), (3,1), (1,3).

Index entries for sequences related to sums of squares

Cf. A000328, A000603, A001182, A008442, A053415, A290081, A349610, A349611.

Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^2/(4 (1 - x)), {x, 0, n^2}], {n, 0, 55}]

a(n) = [x^(n^2)] theta_2(x^4)^2 / (4 * (1 - x)).
a(n) = Sum_{k=0..n^2} A290081(k).
a(n) = A053415(n) / 4.

cp's OEIS Frontend

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

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A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

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A048149 Array T read by diagonals: T(i,j) = number of pairs (h,k) with h^2+k^2 <= i^2+j^2, h>=0, k >= 0.

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A036702 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b<=a.

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A228233 Number of Gaussian primes of norm less than or equal to n in the first quadrant.

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Mathematica

A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

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A255238 Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m in the first quadrant.

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A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.