A000328 -id:A000328 - cp's OEIS frontend

A093836 Numerator of A000328(n)/n^2, where A000328(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.

5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 49, 529, 613, 709, 797, 53, 1009, 1129, 1257, 1373, 1517, 1653, 1793, 1961, 2121, 763, 2453, 2629, 2821, 3001, 3209, 3409, 3625, 3853, 1351, 4293, 4513, 4777, 201, 5261, 5525, 5789, 6077, 6361, 6625

Offset: 1

Eric W. Weisstein, Apr 17 2004

First differs from A000328 at n=12: A000328(12) = 441, a(12) = 49.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gauss's Circle Problem

Cf. A093837 (denominators), A000328, A093832.

N:= 100: # to get a(1) to a(N)
B:= Array(1..N);
for i from 0 to N do
  for j from i while i^2 + j^2 <= N^2 do
     v:= ceil(sqrt(i^2+j^2));
     if [i,j] = [0,0] then m:= 1; v:= 1
     elif i=0 or i=j then m:= 4
     else m:= 8
     fi;
     B[v]:= B[v]+m;
  od
od:
A000328:= ListTools:-PartialSums(convert(B,list)):
seq(numer(A000328[n]/n^2),n=1..N); # Robert Israel, May 28 2015

Definition edited by Robert Israel, May 28 2015

A381018 a(n) is the number of primes in A000328 for r <= n.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 5, 6, 6, 7, 7, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 22, 22, 23, 23, 23

Offset: 1

Michel Marcus, Feb 12 2025

nonn

Number of Gauss circle primes for r <= n.

Thomas Ehrenborg, Gauss Circle Primes, arXiv:2502.06804 [math.GM], 2025. See Table 2 p. 4.

Cf. A000328, A000720, A381017, A381020.

a(n) = sum(k=0, n, isprime(1 + 4*sum(j=0, k^2\4, k^2\(4*j+1) - k^2\(4*j+3))));

from math import isqrt
from sympy import isprime
def A381018(n): return sum(1 for m in range(1,n+1) if isprime(1+(sum(isqrt(k*((m<<1)-k)) for k in range(1,m+1))<<2))) # Chai Wah Wu, Feb 13 2025

A381017 Prime terms of A000328.

Original entry on oeis.org

5, 13, 29, 113, 149, 197, 317, 613, 709, 797, 1009, 1129, 1373, 3001, 3209, 3853, 4513, 5261, 6361, 7213, 11681, 12853, 15373, 16729, 19577, 20593, 21101, 22133, 25997, 30757, 33317, 38669, 53077, 56401, 65101, 68777, 72533, 73517, 95093, 100621, 108637, 114553, 115781, 118213

Offset: 1

Michel Marcus, Feb 12 2025

nonn

Called Gauss circle primes by Ehrenborg.

Robert Israel, Table of n, a(n) for n = 1..1188
Thomas Ehrenborg, Gauss Circle Primes, arXiv:2502.06804 [math.GM], 2025. See Table 1 p. 3.

Cf. A000040, A000328, A381018.

N:= 200: # for terms in A000328(1..N)
V:= Array(0..N): V[0]:= 1:
for x from 1 to N do
  for y from 0 to x do
    if y = 0 or y = x then m:= 4 else m:= 8 fi;
    s:= ceil(sqrt(x^2+y^2));
    if s > N then break fi;
    V[s]:= V[s] + m
od od:
select(isprime, ListTools:-PartialSums(convert(V,list))); # Robert Israel, May 27 2025

select(isprime, vector(200, n, 1 + 4*sum(j=0, n^2\4, n^2\(4*j+1) - n^2\(4*j+3))))

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)

A057655 The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.

Original entry on oeis.org

1, 5, 9, 9, 13, 21, 21, 21, 25, 29, 37, 37, 37, 45, 45, 45, 49, 57, 61, 61, 69, 69, 69, 69, 69, 81, 89, 89, 89, 97, 97, 97, 101, 101, 109, 109, 113, 121, 121, 121, 129, 137, 137, 137, 137, 145, 145, 145, 145, 149, 161, 161, 169, 177, 177, 177

Offset: 0

N. J. A. Sloane, Oct 15 2000

			a(0) = 1 (counting origin).
a(1) = 5 since 4 points lie on the circle of radius sqrt(1) + origin.
a(2) = 9 since 4 lattice points lie on the circle w/radius = sqrt(2) (along diagonals) + 4 points inside the circle + origin. - _Wesley Ivan Hurt_, Jan 10 2013

C. Alsina and R. B. Nelsen, Charming Proofs: A Journey Into Elegant Mathematics, Math. Assoc. Amer., 2010, p. 42.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
F. Fricker, Einfuehrung in die Gitterpunktlehre, Birkhäuser, Boston, 1982.
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 5.
E. Kraetzel, Lattice Points, Kluwer, Dordrecht, 1988.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
W. Sierpiński, Elementary Theory of Numbers, Elsevier, North-Holland, 1988.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 245-246.

T. D. Noe, Table of n, a(n) for n = 0..1000
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
F. Richman, Counting Gaussian integers in a disk
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.

Partial sums of A004018. Cf. A014198, A057656, A057961, A057962, A122510. For another version see A000328.

Cf. A038589 (for hexagonal lattice).

a057655 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
R:= Array(0..N):
for a from 0 to floor(sqrt(N)) do
  for b from 0 to floor(sqrt(N-a^2)) do
    r:= a^2 + b^2;
    R[r]:= R[r] + (2 - charfcn[0](a))*(2 - charfcn[0](b));
  od
od:
convert(map(round,Statistics:-CumulativeSum(R)),list); # Robert Israel, Sep 29 2014

f[n_] := 1 + 4Sum[ Floor@ Sqrt[n - k^2], {k, 0, Sqrt[n]}]; Table[ f[n], {n, 0, 60}] (* Robert G. Wilson v, Jun 16 2006 *)
Accumulate[ SquaresR[2, Range[0, 55]]] (* Jean-François Alcover, Feb 24 2012 *)
CoefficientList[Series[EllipticTheta[3,0,x]^2/(1-x), {x, 0, 100}], x] (* Vaclav Kotesovec, Sep 29 2014 after Robert Israel *)

a(n)=sum(x=-n,n,sum(y=-n,n,if((sign(x^2+y^2-n)+1)*sign(x^2+y^2-n),0,1)))

a(n)=1+4*sum(k=0,sqrtint(n), sqrtint(n-k^2) ); /* Benoit Cloitre, Oct 08 2012 */

from math import isqrt
def A057655(n): return 1+(sum(isqrt(n-k**2) for k in range(isqrt(n)+1))<<2) # Chai Wah Wu, Jul 31 2023

a(n) = 1 + 4*{ [n/1] - [n/3] + [n/5] - [n/7] + ... }. - Gauss
a(n) = 1 + 4*Sum_{ k = 0 .. [sqrt(n)] } [ sqrt(n-k^2) ]. - Liouville (?)
a(n) - Pi*n = O(sqrt(n)) (Gauss). a(n) - Pi*n = O(n^c), c = 23/73 + epsilon ~ 0.3151 (Huxley). If a(n) - Pi*n = O(n^c) then c > 1/4 (Landau, Hardy). It is conjectured that a(n) - Pi*n = O(n^(1/4 + epsilon)) for all epsilon > 0.
a(n) = A014198(n) + 1.
a(n) = A122510(2,n). - R. J. Mathar, Apr 21 2010
a(n) = 1 + sum((floor(1/(k+1)) + 4 * floor(cos(Pi * sqrt(k))^2) - 4 * floor(cos(Pi * sqrt(k/2))^2) + 8 * sum((floor(cos(Pi * sqrt(i))^2) * floor(cos(Pi * sqrt(k-i))^2)), i = 1..floor(k/2))), k = 1..n). - Wesley Ivan Hurt, Jan 10 2013
G.f.: theta_3(0,x)^2/(1-x) where theta_3 is a Jacobi theta function. - Robert Israel, Sep 29 2014
a(n^2) = A000328(n). - R. J. Mathar, Aug 03 2025

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

			Square array begins:
  1,   1,   1,    1,     1,      1,  ...
  1,   3,   5,    7,     9,     11,  ...
  1,   5,  13,   33,    89,    221,  ...
  1,   7,  29,  123,   425,   1343,  ...
  1,   9,  49,  257,  1281,   5913,  ...
  1,  11,  81,  515,  3121,  16875,  ...

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, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
Main diagonal gives A302861.
Cf. A000122, A122510, A302996, A302998.

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^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, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, 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=-infinity..infinity} x^(j^2))^k.

A305575 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

Original entry on oeis.org

0, 1, 0, -1, 0, 1, -1, -1, 1, 2, 0, -2, 0, 2, 1, -1, -2, -2, -1, 1, 2, 2, -2, -2, 2, 3, 0, -3, 0, 3, 1, -1, -3, -3, -1, 1, 3, 3, 2, -2, -3, -3, -2, 2, 3, 4, 0, -4, 0, 4, 1, -1, -4, -4, -1, 1, 4, 3, -3, -3, 3, 4, 2, -2, -4, -4, -2, 2, 4, 5, 4, 3, 0, -3, -4, -5, -4, -3, 0, 3, 4, 5, 1, -1

Offset: 0

Hugo Pfoertner, Jun 05 2018

Similar to A283307, but with secondary sorting by polar angle.

			The first points (listing [polar angle phi,x,y]) are:
r^2
  0: [0.0*Pi,0,0];
  1: [0.0*Pi,1,0], [0.5*Pi,0,1], [1.0*Pi,-1,0], [1.5*Pi,0,-1];
  2: [0.25*Pi,1,1], [0.75*Pi,-1,1], [1.25*Pi,-1,-1], [1.75*Pi,1,-1];
  4: [0.0*Pi,2,0], [0.5*Pi,0,2], [1.0*Pi,-2,0], [1.5*Pi,0,-2];
  5: [0.148*Pi,2,1], [0.352*Pi,1,2], [0.648*Pi,-1,2], [0.852*Pi,-2,1],
   [1.148*Pi,-2,-1], [1.352*Pi,-1,-2], [1.648*Pi,1,-2], [1.852*Pi,2,-1];
  8: [0.25*Pi,2,2], [0.75*Pi,-2,2], [1.25*Pi,-2,-2], [1.75*Pi,2,-2].

Hugo Pfoertner, Table of n, a(n) for n = 0..17664 (covering range r <= 75).
Hugo Pfoertner, Illustration of A305576 vs A305575.
Rainer Rosenthal, Coloured Circular Visualization of Spiral [A305575,A305576]

For the y-coordinates see A305576.

Cf. A000328, A004018, A283307, A283308, A307014, A307016, A307017, A308080, A308081.

atan2(y,x)=if(x>0,atan(y/x),if(x==0,if(y>0,Pi/2,-Pi/2),if(y>=0,atan(y/x)+Pi,atan(y/x)-Pi)));
angle(x,y)=(atan2(y,x)+2*Pi)%(2*Pi);
{a004018(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))};
xyselect=1; \\ change to 2 for A305576
print1(0,", ");for(s=1,25,my(r=a004018(s));if(r>0,my(v=matrix(r,3),w=vector(r),m=sqrtint(s),L=0);for(i=-m,m,my(k=s-i^2,kk);if(k==0,v[L++,1]=i;v[L,2]=0;v[L,3]=angle(i,0),if(issquare(k),kk=sqrtint(k);forstep(j=-kk,kk,kk+kk,v[L++,1]=i;v[L,2]=j;v[L,3]=angle(i,j)))));p=vecsort(v[,3],,1);for(k=1,L,w[k]=v[p[k],xyselect]);for(k=1,L,print1(w[k],", ")))); \\ Hugo Pfoertner, May 12 2019

A305576 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives y-coordinates.

Original entry on oeis.org

0, 0, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 2, 2, 1, -1, -2, -2, -1, 2, 2, -2, -2, 0, 3, 0, -3, 1, 3, 3, 1, -1, -3, -3, -1, 2, 3, 3, 2, -2, -3, -3, -2, 0, 4, 0, -4, 1, 4, 4, 1, -1, -4, -4, -1, 3, 3, -3, -3, 2, 4, 4, 2, -2, -4, -4, -2, 0, 3, 4, 5, 4, 3, 0, -3, -4, -5, -4, -3, 1

Offset: 0

Hugo Pfoertner, Jun 05 2018

Similar to A283308, but with secondary sorting by polar angle.

			See A305575.

Rainer Rosenthal, Table of n, a(n) for n = 0..17664, covering range r <= 75 (terms 0..7844 from Hugo Pfoertner).

For the x-coordinates see A305575.

Cf. A000328, A004018, A283307, A283308, A307014, A307016, A307017, A308080, A308081.

PARI
```
\\ Provided in A305575.
```

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

A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.

Original entry on oeis.org

0, 1, 9, 25, 45, 69, 109, 145, 193, 249, 305, 373, 437, 517, 609, 697, 793, 889, 1005, 1125, 1245, 1369, 1513, 1649, 1789, 1941, 2109, 2285, 2449, 2617, 2809, 2997, 3205, 3405, 3613, 3841, 4049, 4281, 4509, 4765, 5013, 5249, 5521, 5785, 6073, 6349, 6621

Offset: 0

Jostein Trondal (jostein.trondal(AT)protech.no)

			a(3)=25 from the points of shapes 00 (1), 10 (4), 11 (4), 20 (4), 21 (8), 22 (4).

T. D. Noe, Table of n, a(n) for n = 0..1000
Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. See p. 18.

Changing "<" to "<=" in the definition gives A000328.

a051132 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 - 1}], {n, 0, 46}]
a[0]=0;a[n_]:=4*n-3+4Sum[Ceiling[Sqrt[n^2-i^2]]-1,{i,n-1}];Array[a,47,0] (* Giorgos Kalogeropoulos, May 20 2025 *)

from math import isqrt
def A051132(n): return 1+(sum(isqrt(k*((n<<1)-k)-1) for k in range(1,n+1))<<2) if n else 0 # Chai Wah Wu, Feb 12 2025

a(n) = A000328(n) - A046109(n). - Reinhard Zumkeller, Jan 23 2012
Limit_{n->oo} a(n)/n^2 = Pi. - Chai Wah Wu, Feb 12 2025
a(n) = 4*n - 3 + 4 Sum_{i=1..n-1} ceiling(sqrt(n^2 - i^2)) - 1, for n > 0 (see Zhao). - Giorgos Kalogeropoulos, May 20 2025

More terms from James Sellers

cp's OEIS Frontend

A093836 Numerator of A000328(n)/n^2, where A000328(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.

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A381018 a(n) is the number of primes in A000328 for r <= n.

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A381017 Prime terms of A000328.

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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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A057655 The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.

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A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

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A305575 List points (x,y) having integer coordinates, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives x-coordinates.

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