A000328 -id:A000328 - cp's OEIS frontend

A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

1, 3, 13, 123, 1281, 16875, 252673, 4031123, 70554353, 1318315075, 26107328109, 549772933959, 12147113355505, 280978137279483, 6780378828922333, 169829490474843659, 4409771551548703649, 118361723203178140163, 3277041835527134201777, 93455465161026267454527

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius n.

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

Main diagonal of A302997.

Cf. A000122, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416, A122510, A232173, A302860.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n^2}], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n^2}], {n, 0, 19}]

a(n) = A122510(n,n^2).

A303644 a(n) is the number of lattice points in a Cartesian grid between a square of side length 2*n, centered at the origin, and its inscribed circle. The sides of the square are parallel to the coordinate axes.

Original entry on oeis.org

0, 0, 0, 4, 4, 12, 24, 32, 40, 48, 68, 92, 100, 120, 136, 168, 192, 220, 244, 268, 312, 336, 376, 420, 444, 484, 524, 576, 624, 664, 724, 764, 820, 868, 912, 992, 1040, 1116, 1156, 1220, 1304, 1368, 1440, 1496, 1564, 1660, 1732, 1816, 1888, 1960, 2032, 2116, 2220, 2308

Offset: 1

Kirill Ustyantsev, Apr 27 2018

nonn

The borders of the square and the circle are not included. Rotating the square by 45 degrees (so that its vertices lie on the coordinate axes) results in sequence A303646 instead.

			For n = 4, we have 4 points with integer coordinates; the point in the first quadrant is at (3,3):
.
                    o . . + . . o
                    . . . + . . .
                    . . . + . . .
                   -+-+-+-+-+-+-+-
                    . . . + . . .
                    . . . + . . .
                    o . . + . . o
.
Similarly, for n = 5, we have 4 points with integer coordinates; the point in the first quadrant is at (4,4):
.
                  o . . . + . . . o
                  . . . . + . . . .
                  . . . . + . . . .
                  . . . . + . . . .
                 -+-+-+-+-+-+-+-+-+-
                  . . . . + . . . .
                  . . . . + . . . .
                  . . . . + . . . .
                  o . . . + . . . o
.
For n = 6, we have 12 points, of which the 3 points in the first quadrant are at (4,5), (5,4), and (5,5):
.
                o o . . . + . . . o o
                o . . . . + . . . . o
                . . . . . + . . . . .
                . . . . . + . . . . .
                . . . . . + . . . . .
               -+-+-+-+-+-+-+-+-+-+-+-
                . . . . . + . . . . .
                . . . . . + . . . . .
                . . . . . + . . . . .
                o . . . . + . . . . o
                o o . . . + . . . o o

Kirill Ustyantsev, illustrated example

Cf. A000328, A016754, A302829, A303642, A303646.

a(n) = sum(x=-n+1, n-1, sum(y=-n+1, n-1, (x^2+y^2) > n^2)); \\ Michel Marcus, May 22 2018

import math
for n in range(1, 100):
    count = 0
    for x in range(1, n):
        for y in range(1, n):
            if x * x + y * y > n * n and x < n and y < n:
                count = count + 1
    print(4 * count, end=", ")

a(n) = A016754(n-1) - A000328(n) - 4.

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

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).

A175341 Number of coprime pairs (x,y) with x^2+y^2 <= n^2.

Original entry on oeis.org

0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192, 224, 264, 312, 384, 440, 480, 544, 616, 672, 768, 832, 928, 1000, 1112, 1192, 1280, 1384, 1488, 1584, 1704, 1816, 1960, 2072, 2224, 2344, 2480, 2600, 2752, 2912, 3064, 3184, 3360, 3480, 3696, 3856, 4016, 4176

Offset: 0

R. J. Mathar, Apr 16 2010

nonn

			a(2) = 8 counts (x,y) = (-1,-1), (-1,0), (-1,1), (0,-1), (0,1), (1,-1), (1,0) and (1,1).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Gauss circle problem
J. Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), 69.
W. G. Zhai and X.D. Cao, On the number of coprime integer pairs within a circle, Acta Arith. 90 (1999), 1.

Cf. A000328, A176562, A304651.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
b[n_] := b[n] = If[n == 0, 0, b[n-1] + 4 a89[n]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 02 2023 *)

a(n) = 4*A176562(n). - R. J. Mathar, May 07 2010

a(n) = A304651(n^2). - Seiichi Manyama, May 26 2018

A292276 a(n) = number of vertices of the convex hull of the set of points of norm <= n^2 in square lattice.

Original entry on oeis.org

1, 4, 4, 8, 12, 12, 12, 12, 20, 12, 20, 20, 20, 20, 20, 20, 20, 24, 28, 20, 20, 20, 36, 36, 28, 20, 36, 36, 36, 36, 28, 36, 36, 36, 44, 36, 36, 36, 36, 52, 44, 36, 36, 36, 52, 44, 52, 52, 44, 52, 44, 60, 52, 44, 52, 44, 52, 52, 52, 52, 60, 52, 52, 52, 68, 44

Offset: 0

Rémy Sigrist, Sep 13 2017

nonn

The convex hull of a finite point set in dimension 2, say S, forms a convex polygon whose vertices are in S.
For any n >= 0, A000328(n) gives the number of elements of the set of points of norm <= n^2 in square lattice.
For symmetry reasons, a(n) is a multiple of 4 for any n > 0.

			See Links section.

Rémy Sigrist, C++ program for A292276
Rémy Sigrist, Illustration of the convex hulls for n=0..10
Wikipedia, Convex hull of a finite point set

Cf. A000328, A046109.

A295344 Maximum number of lattice points inside and on a circle of radius n.

Original entry on oeis.org

1, 5, 14, 32, 52, 81, 116, 157, 208, 258, 319, 384, 457, 540, 623, 716, 812, 914, 1025, 1142, 1268, 1396, 1528, 1669, 1816, 1976, 2131, 2300, 2472, 2650, 2836, 3028, 3228, 3436, 3644, 3859, 4080, 4314, 4548, 4792, 5038, 5289, 5555, 5818, 6092, 6376, 6668, 6952

Offset: 0

Arkadiusz Wesolowski, Nov 20 2017

nonn

Maximum number of lattice points (i.e., points with integer coordinates) in the plane that can be covered by a circle of radius n.
a(n) >= A000328(n).
Conjecture: sequence contains infinitely many terms that are divisible by 4.

			For a circle centered at the point (x, y) = (1/2, 1/4) with radius 2, there are 14 lattice points inside and on the circle.
.
.     Center             # Pts in/
.    x      y    Radius  on circle
.  -----  -----  ------  ---------
.    0      0       1         5
.   1/2    1/4      2        14
.   1/2    1/2      3        32
.   1/2    1/2      4        52
.    0      0       5        81
.   1/2    1/3      6       116
.   2/5    1/5      7       157
.   1/2    1/2      8       208
.   1/2    2/9      9       258
.  20/47  19/56    10       319
.   1/2    1/2     11       384
.  11/23   7/20    12       457
.   1/2    1/2     13       540
.  10/21   3/13    14       623
.   1/2    1/2     15       716
.   1/2    1/2     16       812
.   2/5    2/5     17       914
.   3/8    5/14    18      1025
.   1/2    1/6     19      1142
.   9/19   8/17    20      1268

B. R. Srinivasan, Lattice Points in a Circle, Proc. Nat. Inst. Sci. India, Part A, 29 (1963), pp. 332-346.

Cf. A000328, A123690, A291259.

L=List([]); for(n=0, 47, if(n>0, j=5, j=1); g=0; h=0; f=ceil(Pi*n^2); for(d=2, floor(f/2), for(c=1, floor(d/2), if(gcd(c, d)==1, for(e=d, d+1, if(e/f<=1/2, a=c/d; b=e/f; if(a+b>=1/2, t=0; for(x=-n, n+1, for(y=-n, n+1, z=(a-x)^2+(b-y)^2; if(z<=n^2,t++))); if(t>j, j=t; if(a>=b, g=a; h=b, g=b; h=a)))))))); print("a("n") = "j", the center of the circle is at point ("g", "h")."); listput(L, j)); print(); print(Vec(L));

a(n) = Pi*n^2 + O(n), as n goes to infinity.

a(n) = A123690(2*n) for n >= 1.

a(10) corrected by Giovanni Resta, Nov 24 2017

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.

A357576 Half area of the convex hull of {(x,y)| x,y integers and x^2 + y^2 < n^2}.

Original entry on oeis.org

0, 2, 8, 17, 28, 46, 63, 87, 112, 142, 173, 204, 244, 287, 333, 378, 428, 485, 540, 602, 661, 737, 802, 869, 947, 1030, 1118, 1197, 1278, 1378, 1469, 1575, 1670, 1776, 1889, 1990, 2108, 2219, 2353, 2472, 2587, 2723, 2854, 3002, 3135, 3275, 3424, 3563, 3721

Offset: 1

Gerhard Kirchner, Oct 05 2022

nonn

a(n) is odd if there is an edge connecting two vertices (x,y) and (y,x), x > y > 0, such that x-y is odd. Otherwise, a(n) is even. a(n)/n^2 is not monotonous but tends to Pi/2. The convex hull has four symmetry axes: x=0, y=0, y=x, y=-x. Therefore it is sufficient to find the least area of a quarter polygon (multiplied by 2). The half area is an integer because the area of any convex polygon whose vertex coordinates are integers is a multiple of 1/2.

			For n=4: 5+6+6 = 17 square units -> a(4)=17.
     _______
   /|_|_|_|_|\  5
  |_|_|_|_|_|_| 6
  |_|_|_|_|_|_| 6

Gerhard Kirchner, Examples for n <= 13.

Cf. A000328, A004144, A292276, A357575.

block(nmax: 40, a: makelist(0,i,1,nmax), a[1]:0,
for n from 2 thru nmax do
  (x0:0, y0:n, xa:0, ya:n, m1:0, m0:2, ar:0,
    while xa

from math import isqrt
from sympy import convex_hull
def A357576(n): return 0 if n == 1 else int(2*convex_hull(*[(0,0),(n-1,0)]+[(x,isqrt((n-x)*(n+x)-1)) for x in range(n)]).area) # Chai Wah Wu, Oct 23 2022

a(n) = A357575(n) - 2*floor(sqrt(2*n-1)) if n is a nonhypotenuse number (A004144).

A036693 Number of Gaussian integers z = a + bi satisfying n-1 < |z| <= n.

Original entry on oeis.org

1, 4, 8, 16, 20, 32, 32, 36, 48, 56, 64, 60, 64, 88, 84, 96, 88, 104, 108, 120, 128, 116, 144, 136, 140, 168, 160, 168, 164, 176, 192, 180, 208, 200, 216, 228, 200, 240, 220, 264, 248, 236, 264, 264, 288, 284, 264, 296, 292, 312

Offset: 0

Clark Kimberling

nonn

			a(10^2) = 660, a(10^3) = 6392, a(10^4) = 62952, a(10^5) = 628520, a(10^6) = 6281404. - _Reinhard Zumkeller_, Jan 13 2002

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for Gaussian integers and primes

Cf. A000328.

[#[: x in [-n..n], y in [-n..n]| n-1 lt r and r  le n where r is Sqrt(x^2+ y^2)]: n in [0..50]]; // Marius A. Burtea, Feb 18 2020

def A036693(n):
    if n == 0: return 1
    Range = lambda n: ((i, j) for i in (-n..n) for j in (-n..n))
    return sum(1 for (j, k) in Range(n) if (n-1)^2 < j^2 + k^2 <= n^2)
print([A036693(n) for n in range(20)]) # Peter Luschny, Mar 27 2020

From Reinhard Zumkeller, Jan 13 2002: (Start)
a(n)/n ~ 2*Pi.
a(n) = A000328(n)-A000328(n-1). (End)

cp's OEIS Frontend

A302861 a(n) = [x^(n^2)] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

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A303644 a(n) is the number of lattice points in a Cartesian grid between a square of side length 2*n, centered at the origin, and its inscribed circle. The sides of the square are parallel to the coordinate axes.

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

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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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Mathematica

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A175341 Number of coprime pairs (x,y) with x^2+y^2 <= n^2.

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Mathematica

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A292276 a(n) = number of vertices of the convex hull of the set of points of norm <= n^2 in square lattice.

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A295344 Maximum number of lattice points inside and on a circle of radius n.

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

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