A051132 -id:A051132 - 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

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

Original entry on oeis.org

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36

Offset: 0

Eric W. Weisstein

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012

The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.

Cf. A000328, A051132, A009177.

a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015

Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)

a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, 2*f[i,2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017

a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

a(n) = A000328(n) - A051132(n).
a(n) = 8*A046080(n) + 4 for n > 0.
a(n) = A004018(n^2).
From Jean-Christophe Hervé, Dec 01 2013: (Start)
a(A084647(k)) = 28.
a(A084648(k)) = 36.
a(A084649(k)) = 44. (End)
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017
a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018
From Hugo Pfoertner, Sep 21 2023: (Start)
a(n) = 8*A063014(n) - 4 for n > 0.
a(n) = 4*A256452(n) for n > 0. (End)

A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n.

Original entry on oeis.org

0, 1, 9, 25, 45, 69, 108, 145, 193, 248, 305, 373, 437, 517, 608, 697, 793, 889, 1005, 1124, 1245, 1369, 1510, 1649, 1789, 1941, 2109, 2278, 2449, 2617, 2809, 2997, 3202, 3405, 3613, 3834, 4049, 4281, 4509, 4762, 5013, 5249, 5521, 5785, 6068, 6348, 6621, 6917

Offset: 0

Andres Cicuttin, Aug 21 2017

nonn

Due to the symmetry and periodicity of the square lattice it is sufficient to explore possible circles with center belonging to the triangle with vertices (0,0), (1/2,0), and (1/2,1/2).

The different regions for the centers producing constant numbers of lattice points inside circles of radius n seem to become very complex and irregular as n increases (see density plots in Links).

			From _Arkadiusz Wesolowski_, Dec 18 2017 [Corrected by _Andrey Zabolotskiy_, Feb 19 2018]: (Start)
For a circle centered at the point (x, y) = (1/2, 0) with radius 6, there are 108 lattice points inside the circle.
Possible (but not unique) choices for the centers of the circles for radii up to 20 are given below.
.
.  Poss. center          Points in
.    x      y    Radius  the circle
.  -----  -----  ------  ----------
.    0      0       1         1
.    0      0       2         9
.    0      0       3        25
.    0      0       4        45
.    0      0       5        69
.   1/2     0       6       108
.    0      0       7       145
.    0      0       8       193
.   1/5     0       9       248
.    0      0      10       305
.    0      0      11       373
.    0      0      12       437
.    0      0      13       517
.   1/4     0      14       608
.    0      0      15       697
.    0      0      16       793
.    0      0      17       889
.    0      0      18      1005
.   1/2    1/2     19      1124
.    0      0      20      1245
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..125
Andres Cicuttin, Plots of regions for centers of circles of several radii n enclosing constant number of lattice points
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A046109, A051132, A295344.

(* A291259: Minimum number of points of the square lattice falling strictly inside a circle of radius n. *)
(* The three vertices of the Explorative Triangle (ET) *)
P1={0,0}; P2={1/2,0}; P3={1/2,1/2};
dd2=SquaredEuclideanDistance;
(* candidatePointQ[p,n] gives True if "p" is a candidate point, and False otherwise. A candidate point is a point belonging to a circle of radius "n" with center in the ET *)
candidatePointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]>=n^2>=Min[dds]];
(* Check if point "p" falls inside any circle with radius "n" and center in the ET *)
innerPointQ[p_,n_] := With[{dds={dd2[p,P1],dd2[p,P2],dd2[p,P3]}}, Max[dds]Andres Cicuttin & Andrey Zabolotskiy, Nov 14 2017 *)

a(n) ~ Pi*n^2.

a(n) <= A051132(n). - Joerg Arndt, Oct 03 2017

More terms from Andrey Zabolotskiy, Nov 17 2017

A077770 Number of ordered pairs of integers (x,y) with n^2 < x^2 + y^2 < (n+1)^2; number of lattice points between circles of radii n and n+1.

Original entry on oeis.org

0, 4, 12, 16, 20, 28, 32, 44, 52, 52, 56, 60, 76, 80, 84, 84, 92, 104, 116, 116, 112, 140, 132, 136, 148, 148, 164, 160, 164, 180, 176, 204, 196, 204, 216, 196, 228, 216, 252, 236, 224, 260, 260, 284, 272, 260, 292, 288, 308, 300, 316, 312, 300, 332, 320, 364

Offset: 0

T. D. Noe, Nov 20 2002

nonn

Note that 2*A077768(n)-a(n)/4 is the characteristic sequence for the Beatty sequence A001951(n).

Eric Weisstein's World of Mathematics, Gauss's Circle Problem

Cf. A000328, A001951, A051132, A077768.

Table[Sum[SquaresR[2, k], {k, n^2 + 1, (n + 1)^2 - 1}], {n, 0, 100}]

a(n) = A051132(n+1) - A000328(n)

A047077 Number of pairs of integers (x,y) with n^2 <= x^2+y^2 <= (n+1)^2.

Original entry on oeis.org

5, 12, 20, 24, 36, 44, 40, 52, 60, 68, 72, 68, 92, 96, 100, 100, 108, 120, 124, 132, 128, 148, 140, 144, 172, 180, 180, 168, 180, 204, 192, 212, 204, 220, 240, 212, 244, 232, 268, 260, 248, 276, 268, 292, 288, 276, 300, 296, 316, 324, 348, 336, 324, 348, 336

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 26 2000

Number of integer Cartesian grid points covered by a ring around the origin with width 1 and outer radius n. Also, number of Gaussian integers z=a+bi satisfying n-1 <= |z| <= n. - Ralf Stephan, Nov 28 2013

Cf. A000328, A051132, A077770.

a[n_] := Sum[SquaresR[2, k], {k, n^2, (n+1)^2}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Oct 15 2012 *)

a(n) = A000328(n+1) - A051132(n).

A053456 Open disk numbers (version 1): a(n) is the number of points (i,j), i,j, integers, contained in an open disk of diameter n, centered at (0,0).

Original entry on oeis.org

0, 1, 1, 9, 9, 21, 25, 37, 45, 69, 69, 97, 109, 137, 145, 177, 193, 225, 249, 293, 305, 349, 373, 421, 437, 489, 517, 577, 609, 665, 697, 749, 793, 861, 889, 973, 1005, 1085, 1125, 1201, 1245, 1313, 1369, 1457, 1513, 1597, 1649, 1741, 1789, 1885, 1941

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 13 2000

Cf. A053411, A053457, A053458, A053459.
Bisections: A051132, A036704.
Cf. A000796 (Pi).

a(n)/(n/2)^2->Pi.

A088898 T(n,k) = number of ordered pairs of integers (x,y) with x^2/n^2 + y^2/k^2 < 1, 1<=k<=n; triangular array, read by rows.

Original entry on oeis.org

1, 3, 9, 5, 15, 25, 7, 21, 31, 45, 9, 27, 41, 59, 69, 11, 33, 51, 69, 87, 109, 13, 39, 61, 83, 105, 127, 145, 15, 41, 67, 93, 119, 141, 171, 193, 17, 47, 77, 103, 137, 159, 193, 219, 249, 19, 53, 87, 117, 147, 181, 215, 241, 275, 305, 21, 59, 97, 131, 165, 203

Offset: 1

Reinhard Zumkeller, Oct 21 2003

T(n,k) = number of inner lattice points of an ellipse with semimajor axis = n, semiminor axis = k and center = (0,0).
a(n) = A088897(n) - A088899(n);
T(n,n) = A051132(n).

Eric Weisstein's World of Mathematics, Ellipse

T[1, 1] = 1;
T[n_, k_] := Reduce[x^2/n^2 + y^2/k^2 < 1, {x, y}, Integers] // Length;
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 27 2021 *)

A103225 Number of Gaussian integers z with abs(z) < n and gcd(n,z)=1.

Original entry on oeis.org

1, 4, 24, 24, 44, 48, 144, 96, 224, 96, 372, 192, 444, 304, 404, 392, 792, 448, 1124, 408, 1200, 752, 1648, 808, 1240, 896, 2036, 1200, 2440, 800, 2996, 1600, 3008, 1592, 2404, 1808, 4056, 2256, 3616, 1600, 4992, 2400, 5784, 3008, 3604, 3304, 6916, 3224, 7376

Offset: 1

T. D. Noe, Jan 26 2005

This sequence is much like the usual totient function. That is, it gives the number of Gaussian integers that are relatively prime to n and whose modulus is less than n. When n is a Gaussian prime, A002145, then a(n) = A051132(n)-1.

Four of the dominant lines of the plot appear to align to k(i)*Pi*n^2, with k(i) = 1, 8/9, 1/2, and 4/9. Conjecture: a(n) < Pi*n^2. - Bill McEachen, Aug 14 2025

			a(2)=4 because 1, -1, i and -i are relatively prime to 2 and have modulus less than 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)

Cf. A002145, A051132, A103222, A103223, A103224.

Table[cnt=0; Do[z=a+ b*I; If[Abs[z]

A387220 Arithmetic mean of number of lattice points strictly inside circle of radius n centered on origin, and number of points not outside that circle.

Original entry on oeis.org

3, 11, 27, 47, 75, 111, 147, 195, 251, 311, 375, 439, 523, 611, 703, 795, 895, 1007, 1127, 1251, 1371, 1515, 1651, 1791, 1951, 2115, 2287, 2451, 2623, 2815, 2999, 3207, 3407, 3619, 3847, 4051, 4287, 4511, 4771, 5019, 5255, 5523, 5787, 6075, 6355, 6623, 6919, 7211

Offset: 1

Lorraine Lee, Aug 22 2025

a(n) is the number of integer values (x, y) strictly inside the circle x^2+y^2=n^2, plus half the number of such lattice points that are part of the perimeter of that circle.

All terms are odd. - Chai Wah Wu, Aug 23 2025

			The unit circle has 1 lattice point strictly inside it and 5 lattice points not outside it. Halfway between 1 and 5 is 3, so a(1) = 3.

Average of A051132 and A000328.

a(n) = {4*n -1 + 2*sum(k=1, n-1, my(t=n^2-k^2); 2*sqrtint(t)-issquare(t))} \\ Andrew Howroyd, Aug 22 2025

from math import isqrt
def A387220(n): return 1+(sum(isqrt(m:=k*((n<<1)-k))+isqrt(m-1) for k in range(1,n+1))<<1) # Chai Wah Wu, Aug 23 2025

a(n) = (A051132(n) + A000328(n))/2.

a(n) = A256465(n^2). - R. J. Mathar, Aug 26 2025

A304035 a(n) is the number of lattice points inside a square bounded by the lines x=-n/sqrt(2), x=n/sqrt(2), y=-n/sqrt(2), y=n/sqrt(2).

Original entry on oeis.org

1, 9, 25, 25, 49, 81, 81, 121, 169, 225, 225, 289, 361, 361, 441, 529, 625, 625, 729, 841, 841, 961, 1089, 1089, 1225, 1369, 1521, 1521, 1681, 1849, 1849, 2025, 2209, 2401, 2401, 2601, 2809, 2809, 3025, 3249, 3249, 3481, 3721, 3969, 3969, 4225, 4489, 4489, 4761, 5041, 5329, 5329, 5625, 5929, 5929

Offset: 1

Kirill Ustyantsev, May 05 2018

nonn

If we calculate the first difference of this sequence and then substitute nonzero numbers as 1, we get exactly A080764.
If we include boundary points of the squares we get same sequence (obviously).
Duplicates appear at 4, 7, 11, 14, 18, 21, 24, 28, 31, 35, 38, 41, 45, 48, 52, 55 (= A083051 ?). - Robert G. Wilson v, Jun 20 2018

Cf. A080764, A049472, A051132, A303642.

a(n) = sum(x=-n, n, sum(y=-n, n, ((2*x^2 < n^2) && (2*y^2 < n^2)))); \\ Michel Marcus, May 22 2018

import math
for n in range (1, 100):
 count=0
 for x in range (-n, n):
  for y in range (-n, n):
   if ((2*x*x < n*n) and (2*y*y < n*n)):
    count=count+1
 print(count)

a(n) = A051132(n) - A303642(n).

cp's OEIS Frontend

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

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A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

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A291259 Minimum number of points of the square lattice falling strictly inside a circle of radius n.

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A077770 Number of ordered pairs of integers (x,y) with n^2 < x^2 + y^2 < (n+1)^2; number of lattice points between circles of radii n and n+1.

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A047077 Number of pairs of integers (x,y) with n^2 <= x^2+y^2 <= (n+1)^2.

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A053456 Open disk numbers (version 1): a(n) is the number of points (i,j), i,j, integers, contained in an open disk of diameter n, centered at (0,0).

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A088898 T(n,k) = number of ordered pairs of integers (x,y) with x^2/n^2 + y^2/k^2 < 1, 1<=k<=n; triangular array, read by rows.

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A103225 Number of Gaussian integers z with abs(z) < n and gcd(n,z)=1.