A001182 -id:A001182 - cp's OEIS frontend

A119677 a(n) is the number of complete squares that fit inside the circle with radius n, drawn on squared paper at (0, 0).

0, 0, 4, 16, 32, 60, 88, 120, 164, 216, 276, 332, 392, 476, 556, 648, 732, 832, 936, 1052, 1176, 1288, 1428, 1560, 1696, 1860, 2016, 2180, 2340, 2512, 2700, 2876, 3080, 3276, 3488, 3712, 3908, 4144, 4360, 4620, 4864, 5096, 5356, 5616, 5900

Offset: 0

Tomas Kyselica (Tomas.Kyselica(AT)gmail.com), Jul 29 2006

nonn

David Dewan, Table of n, a(n) for n = 0..10000
David Dewan, Drawings for n=1..12

Cf. A001182.

a[n_] := 4 Sum[Floor[Sqrt[n^2 - k^2]], {k, n - 1}];
Array[a, 45, 0]  (* David Dewan, May 27 2024 *)

from math import isqrt
def A119677(n): return sum(isqrt(k*((n<<1)-k)) for k in range(1,n))<<2 # Chai Wah Wu, Jul 18 2024

a(n) = 4 * A001182(n).

Limit_{n->oo} a(n)/(n^2) = Pi.

More terms from Don Reble, Aug 01 2006

Name clarified by Kirill Zolotuskiy, Apr 15 2023

A136483 Number of unit square lattice cells inside quadrant of origin-centered circle of diameter n.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 8, 13, 15, 19, 22, 28, 30, 37, 41, 48, 54, 64, 69, 77, 83, 94, 98, 110, 119, 131, 139, 152, 162, 172, 183, 199, 208, 226, 234, 253, 263, 281, 294, 308, 322, 343, 357, 377, 390, 412, 424, 447, 465, 488, 504, 528, 545, 567, 585, 612, 628, 654

Offset: 1

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

			a(5) = 3 because a circle of radius 5/2 in the first quadrant encloses (2,1), (1,1), (1,2).

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Alternating merge of A136484 and A001182.

Cf. A019683, A136485, A136513.

A136483:= func< n | n eq 1 select 0 else (&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136483(n): n in [1..100]]; // G. C. Greubel, Jul 28 2023

Table[Sum[Floor[Sqrt[(n/2)^2 -k^2]], {k,Floor[n/2]}], {n,100}]

a(n) = sum(k=1, n\2, sqrtint((n/2)^2 - k^2)); \\ Michel Marcus, Jul 28 2023

def A136483(n): return sum(isqrt((n/2)^2-j^2) for j in range(1,(n//2)+1))
[A136483(n) for n in range(1,101)] # G. C. Greubel, Jul 28 2023

a(n) = Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).
a(n) = (1/4) * A136485(n) = (1/2) * A136513(n).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (4 * (1 - x)). - Ilya Gutkovskiy, Nov 23 2021

A228232 Number of strict Gaussian primes of norm less than or equal to n in the first quadrant.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 13, 17, 19, 23, 29, 31, 35, 41, 43, 49, 57, 63, 69, 75, 83, 89, 93, 99, 109, 117, 123, 133, 141, 149, 157, 167, 175, 187, 197, 207, 215, 225, 233, 239, 253, 267, 273, 287, 297, 309, 319, 335, 351, 361, 369, 385, 403, 415, 425, 439, 453, 465, 481, 495

Offset: 1

Olivier Gérard, Aug 17 2013

nonn

A Gaussian integer is counted if it has a positive real part and a positive imaginary part (first quadrant excluding the axes).

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

Cf. A001182 (number of strict Gaussian integers in the first quadrant).
Cf. A062711 (counts the Gaussian primes on axes also).
Cf. A228233 (version of this sequence including the axes).

nn = 60; t = Select[Flatten[Table[a + b*I, {a, nn}, {b, nn}]], PrimeQ[#, GaussianIntegers -> True] &]; Table[Length[Select[t, Abs[#] <= n &]], {n, nn}] (* T. D. Noe, Aug 19 2013 *)

A136514 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of radius n.

Original entry on oeis.org

0, 2, 8, 16, 30, 44, 60, 82, 108, 138, 166, 196, 238, 278, 324, 366, 416, 468, 526, 588, 644, 714, 780, 848, 930, 1008, 1090, 1170, 1256, 1350, 1438, 1540, 1638, 1744, 1856, 1954, 2072, 2180, 2310, 2432, 2548, 2678, 2808, 2950, 3090, 3220, 3366, 3510, 3664

Offset: 1

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

			a(2) = 2 because a circle centered at the origin and of radius 2 encloses (-1,1) and (1,1) in the upper half plane.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001182, A136513.

A136514:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt(n^2-j^2)): j in [1..n-1]]) >;
[A136514(n): n in [1..100]]; // G. C. Greubel, Jul 27 2023

Table[2*Sum[Floor[Sqrt[n^2 -k^2]], {k,n-1}], {n,100}]

a(n) = 2*sum(k=1, n-1, sqrtint(n^2-k^2)); \\ Michel Marcus, Jul 27 2023

def A136514(n): return 2*sum(isqrt(n^2-k^2) for k in range(1,n))
[A136514(n) for n in range(1,101)] # G. C. Greubel, Jul 27 2023

Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
a(n) = 2 * Sum_{k=1..n-1} floor(sqrt(n^2 - k^2)).
a(n) = A136513(2*n).
a(n) = 2*A001182(n). - R. J. Mathar, Jan 10 2008

A281795 Number of unit squares (partially) covered by a disk of radius n centered at the origin.

Original entry on oeis.org

0, 4, 16, 36, 60, 88, 132, 172, 224, 284, 344, 416, 484, 568, 664, 756, 856, 956, 1076, 1200, 1324, 1452, 1600, 1740, 1884, 2040, 2212, 2392, 2560, 2732, 2928, 3120, 3332, 3536, 3748, 3980, 4192, 4428, 4660, 4920, 5172, 5412, 5688, 5956, 6248, 6528, 6804, 7104, 7400, 7716

Offset: 0

Orson R. L. Peters, Jan 30 2017

Touching a unit square does not count as covering. E.g., the disk with radius 5 does not cover the unit square with (3, 4) as bottom-left corner.

			a(4) = 4 * 15 = 60 because in the positive quadrant 15 unit squares are covered and the problem is symmetrical. In the bounding box of the circle only the unit squares in the corners are not (partially) covered, so a(4) = 8*8 - 4 = 60.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A001182, A242118.

N:= 100:  # for a(0)..a(N)
V:=Array(0..N):
for i from 0 to N do
  for j from 0 to i do
    r:= sqrt(i^2 + j^2);
    if r::integer then r:= r+1 else r:= ceil(r) fi;
    if r > N then break fi;
    if i=j then m:= 4 else m:= 8 fi;
    V[r..N]:= V[r..N] +~ m;
od od:
convert(V,list); # Robert Israel, Feb 21 2025

A281795[n_] := 4*Sum[Ceiling[Sqrt[n^2 - k^2]], {k, 0, n-1}];
Array[A281795, 100, 0] (* Paolo Xausa, Feb 21 2025 *)

a = @(n) 4*sum(ceil(sqrt(n.^2-(0:n-1).^2))); % Luis Mendo, Aug 09 2021

a = lambda n: sum(4 for x in range(n) for y in range(n)
                    if x*x + y*y < n*n)

a(n) = 4*A001182(n) + A242118(n). - Andrey Zabolotskiy, Jan 30 2017

a(n) = Sum_{k=0..n-1} 4*ceiling(sqrt(n^2-k^2)). - Luis Mendo, Aug 09 2021

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.

A372847 Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.

Original entry on oeis.org

0, 6, 18, 36, 64, 92, 130, 172, 224, 284, 344, 410, 488, 570, 658, 750, 852, 956, 1072, 1194, 1312, 1450, 1584, 1728, 1882, 2044, 2204, 2372, 2548, 2730, 2916, 3112, 3312, 3520, 3738, 3950, 4184, 4408, 4656, 4900, 5146, 5402, 5670, 5942, 6222, 6492, 6784, 7080, 7382, 7700

Offset: 1

David Dewan, May 14 2024

nonn

Always has an even number of rows (2*n-2) and each row may have an odd or even number of squares.

Symmetrical about the horizontal and vertical axes.

			For n=4
row 1:   5 squares
row 2:   6 squares
row 3:   7 squares
row 4:   7 squares
row 5:   6 squares
row 6:   5 squares
Total = 36

David Dewan, Table of n, a(n) for n = 1..10000
David Dewan, Drawings for n=1..12.

Cf. A136485 (by diameter), A001182 (within quadrant), A136483 (quadrant by diameter), A119677 (even number of rows with even number of squares in each), A125228 (odd number of rows with maximal squares per row), A341198 (points rather than squares).

a[n_]:=2 Sum[Floor[2 Sqrt[n^2 - k^2]], {k,n-1}]; Array[a,50]

a(n) = 2*Sum_{k=1..n-1} floor(2*sqrt(n^2 - k^2)).

A120883 (1/4)number of lattice points with odd indices in a square lattice inside a circle around the origin with radius 2n.

Original entry on oeis.org

0, 1, 3, 8, 13, 20, 28, 39, 52, 64, 79, 96, 112, 135, 154, 179, 203, 228, 255, 281, 316, 349, 382, 416, 451, 494, 532, 573, 618, 661, 707, 756, 807, 859, 910, 963, 1015, 1076, 1137, 1198, 1256, 1321, 1386, 1452, 1523, 1594, 1667, 1737, 1808, 1889, 1965, 2046, 2123

Offset: 0

Hugo Pfoertner, Jul 12 2006

nonn

Lim_{n->infinity} a(n)/n^2 = Pi/4.

a(n) gives the number of positive half-points (for example, 1/2 and 3/2) inside or on the circle of radius n. - Jon Perry, Nov 04 2012

			a(3)=8 because the 8 lattice points in the first quadrant (x,y) = {(1,1), (1,3), (3,1), (1,5), (5,1), (3,3), (3,5), (5,3)} all satisfy x^2 + y^2 < (2*3)^2.
a(3)=8 because (1/2,1/2), (1/2,3/2), (1/2,5/2), (3/2,1/2), (3/2,3/2), (3/2,5/2), (5/2,1/2) and (5/2,3/2) all satisfy x^2 + y^2 <= n^2. - _Jon Perry_, Nov 04 2012

Cf. A001182.

a(n) = Sum_{i=1..n} ceiling(sqrt(n^2 - (i - 1/2)^2) - 1/2). Proof outline: consider an integer grid (i,j), e.g., a pixel image. A positive half circle hull's boundary of radius n contains all points ('pixels') where (i - 1/2)^2 + (j - 1/2)^2 = n^2 => j = f(j) = sqrt(n^2 - (i - 1/2)^2) + 1/2. To obtain the number of elements of the hull's closure without nonpositive points, count by upper Riemann sums with interval length 1: (n) = A(n) = Sum_{i=1..n} (ceiling(f(i)) - 1). ('i=1' discards the (0,j) points and '-1' cancels the (i,0) points.) - Johannes Hoentsch, Feb 26 2019

a(0) added by Jon Perry, Nov 04 2012

A097689 Number of prime pairs (p,q) with p^2 + q^2 <= n^2.

Original entry on oeis.org

0, 0, 1, 3, 4, 8, 8, 13, 15, 16, 16, 20, 22, 30, 32, 33, 33, 41, 44, 50, 54, 58, 58, 66, 69, 73, 76, 76, 78, 88, 88, 100, 103, 107, 109, 111, 113, 123, 127, 129, 131, 142, 146, 159, 165, 167, 169, 183, 191, 193, 197, 201, 204, 214, 218, 224, 230, 233, 233, 245, 252, 268

Offset: 1

Reinhard Zumkeller, Sep 21 2004

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..10000

Cf. A001182, A000720.

a[n_] := a[n] = Module[{cnt = 0, p, q}, Do[p = Prime[ip]; q = Prime[iq]; If[p^2 + q^2 <= n^2, If[p == q, cnt++, If[p < q, cnt += 2, 0]]], {ip, PrimePi[n]}, {iq, ip, PrimePi[n]}]; cnt];
Array[a, 100] (* Jean-François Alcover, Nov 18 2021 *)

cp's OEIS Frontend

A119677 a(n) is the number of complete squares that fit inside the circle with radius n, drawn on squared paper at (0, 0).

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A136483 Number of unit square lattice cells inside quadrant of origin-centered circle of diameter n.

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A228232 Number of strict Gaussian primes of norm less than or equal to n in the first quadrant.

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A136514 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of radius n.

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Magma

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A281795 Number of unit squares (partially) covered by a disk of radius n centered at the origin.

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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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A372847 Number of unit squares enclosed by a circle of radius n with an even number of rows and the maximum number of squares in each row.

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A120883 (1/4)number of lattice points with odd indices in a square lattice inside a circle around the origin with radius 2n.