A057655 -id:A057655 - cp's OEIS frontend

A372511 Number of solutions to x^2 + y^2 <= n, where x, y are positive odd integers.

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Index entries for sequences related to sums of squares

Cf. A008442, A057655, A224212, A237526, A290081, A349609, A372512.

nmax = 85; CoefficientList[Series[EllipticTheta[2, 0, x^4]^2/(4 (1 - x)), {x, 0, nmax}], x]

A073092 Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 13, 13, 13, 13, 13, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 27, 27, 28, 29, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 34, 34, 34, 35, 35, 35, 35, 36, 37, 38

Offset: 0

Benoit Cloitre, Aug 18 2002

			0^2 + 0^2, 0^2 + 1^2, 1^2 + 1^2 are less than or equal to 2 hence a(2) = 3.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000161, A001481.

Cf. A057655.

Accumulate @ Table[Length @ PowersRepresentations[n, 2, 2], {n, 0, 100}] (* Amiram Eldar, Mar 08 2020 *)

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

from itertools import count, islice
from math import prod
from sympy import factorint
def A073092_gen(): # generator of terms
    yield (c:=1)
    for n in count(1):
        f = factorint(n)
        c += int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1)
        yield c
A073092_list = list(islice(A073092_gen(),30)) # Chai Wah Wu, Sep 08 2022

a(n) = Sum_{k=0..n} A000161(k).

a(n) is asymptotic to Pi*n/8.

A156790 Number of first quadrant lattice squares inside the circle x^2+y^2=(2^n)^2.

Original entry on oeis.org

0, 1, 8, 41, 183, 770, 3149, 12730, 51209, 205356, 822500, 3292134, 13172634, 52698912, 210812207, 843281848, 3373193506, 13492906143, 53971888157, 215888078393, 863553363881, 3454215553470, 13816866413106, 55267474046659

Offset: 0

Jaume Oliver Lafont, Feb 15 2009

nonn

a(n)/4^n converges to Pi/4 from below.

			Let + represent a square inside the circle and x a square traversed by the circle.
xx
+x a(1)=1
xxx
++xx
+++x
+++x a(2)=8

Wikipedia, Gauss circle problem [From _Jaume Oliver Lafont_, Apr 20 2010]

Cf. A057655.

Cf. A177144. [From Jaume Oliver Lafont, May 03 2010]

a(n)=sum(m=1,2^n-1,floor(sqrt(4^n-m^2)))

a(19) corrected by Sophia Keith, Sep 15 2024

A175373 Partial sums of A175372.

Original entry on oeis.org

1, 5, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 13, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Number of integer pairs (x,y) satisfying x^4+y^4 <= n, any -n <= x,y <=n.

			a(6) = 9 counts (x,y) = (-1,-1), (-1,0), (-1,1), (0,-1), (0,0), (0,1), (1,-1), (1,0) and (1,1).

Cf. A057655, A175363.

A256465 Number of points in a circle of squared radius n, points on the circle counted half.

Original entry on oeis.org

3, 7, 9, 11, 17, 21, 21, 23, 27, 33, 37, 37, 41, 45, 45, 47, 53, 59, 61, 65, 69, 69, 69, 69, 75, 85, 89, 89, 93, 97, 97, 99, 101, 105, 109, 111, 117, 121, 121, 125, 133, 137, 137, 137, 141, 145, 145, 145, 147, 155, 161, 165, 173, 177, 177, 177, 177, 181, 185

Offset: 1

R. J. Mathar, Mar 30 2015

S. W. P. Steen, An expression for the number of lattice-points in a circle, Q. J. Math. 1 (1930) 232-235.

Cf. A057655, A004018.

a(n) = A057655(n) - A004018(n)/2.

A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

Original entry on oeis.org

0, 4, 8, 12, 12, 16, 20, 20, 20, 28, 28, 32, 28, 28, 36, 36, 40, 36, 44, 44, 44, 44, 44, 52, 48, 52, 52, 52, 52, 60, 52, 60, 64, 60, 60, 60, 68, 68, 60, 68, 68, 68, 72, 68, 76, 76, 76, 76, 76, 76, 76, 84, 84, 76, 88, 76, 84, 84, 92, 84, 92

Offset: 1

Scott R. Shannon, Mar 28 2020

nonn

Draw a circle on a 2D square grid centered at the origin with a radius squared equal to the norm of the Gaussian integers A001481(n). See the images in the links. This sequence gives the number of unit cells intersected by the circumference of the circle. Equivalently this is the number of intersections of the circumference with the x and y integer grid lines.

Scott R. Shannon, Illustration for n = 3. The circle has a radius squared of 2, resulting in 8 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 4. The circle has a radius squared of 4, resulting in 12 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 8. The circle has a radius squared of 10, resulting in 20 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 12. The circle has a radius squared of 18, resulting in 32 unit cells intersected/intersection points.
Scott R. Shannon, Illustration for n = 13. The circle has a radius squared of 20, resulting in 28 unit cells intersected/intersection points. This is the first term where the number of intersection points decreases relative to the previous term.
Scott R. Shannon, Illustration for n = 26. The circle has a radius squared of 52, resulting in 52 unit cells intersected/intersection points.
Wikipedia, Gaussian integer.

Cf. A001481, A055025, A057655, A119439, A242118 (a subsequence of this sequence), A234300.

a(n) = 4*A234300(2*(n-1)). - Andrey Zabolotskiy, Feb 22 2025

A363341 Number of positive integers k <= n such that round(n/k) is odd.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 4, 6, 7, 6, 5, 9, 8, 9, 9, 10, 10, 11, 12, 13, 12, 13, 12, 15, 16, 17, 16, 17, 16, 17, 17, 20, 21, 20, 20, 23, 22, 21, 22, 24, 23, 26, 25, 28, 27, 26, 25, 27, 29, 30, 31, 32, 31, 32, 31, 32, 33, 34, 33, 35, 34, 37, 37, 40, 39, 38, 39, 40

Offset: 1

Caleb M. Shor, May 28 2023

nonn

Here round(x) = floor(x + 1/2).

a(n) is related to the number of lattice points in a circle. Let C(x) equal the number of square lattice points in a circle of radius sqrt(x) centered at the origin. Then a(n) = (C(2n) - 4n - 1)/4. (Prop 3.5 in Dent & Shor paper)

			For n=5: round(5/1), round(5/2), round(5/3), round(5/4), round(5/5) = 5, 3, 2, 1, 1 among which 4 are odd so a(5)=4.

Robert Israel, Table of n, a(n) for n = 1..10000
Nicholas Dent and Caleb M. Shor, On residues of rounded shifted fractions with a common numerator, arXiv:2206.15452 [math.NT], 2022.

Cf. A059851 (number of k=1..n such that floor(n/k) is odd).
Cf. A330926 (number of k=1..n such that ceiling(n/k) is odd).
Cf. A057655 (number of lattice points in circle).
Cf. A001826 (d_1), A001842 (d_3), A002654 (d_1-d_3).
Cf. A077024 (n + floor(2n/3) + floor(2n/5) + floor(2n/7) + ...).

f:= proc(n) local k;
   nops(select(k -> floor(n/k + 1/2)::odd, [$1..n]))
end proc:
map(f, [$1..120]); # Robert Israel, Aug 03 2025

a(n) = sum(k=1, n, round(n/k)%2) \\ Andrew Howroyd, May 28 2023

a(n) = n - floor(2n/3) + floor(2n/5) - floor(2n/7) + ...

a(n) = -n + Sum_{k=1..2n} d_1(k) - d_3(k), where d_i(k) is the number of divisors of k that are congruent to i modulo 4.

A356435 a(n) is the minimum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 16, 20, 22, 26, 29, 32, 32, 39, 41, 44, 46, 51, 52, 56, 58, 62, 66, 69, 69, 74, 79, 82, 85, 88, 88, 92, 96, 100, 103, 106, 108, 113, 116, 119, 120, 122, 124, 132, 135, 138, 141, 143, 145, 146, 152, 158, 160, 164, 164, 166, 172, 175, 179, 181, 184, 186, 189, 193, 194, 199

Offset: 0

Bernard Montaron, Aug 07 2022

nonn

a(n) <= A057655(n).

The terms of square index of this sequence are such that a(n^2) = A123689(2n) >= A291259(n), e.g., a(9) = 26 = A123689(6) >= A291259(3) = 25.

			For n = 1 the minimum number of Z x Z lattice points inside the circle is a(1) = 2. The minimum is obtained, for example, with the circle centered at x = 0.1, y = 0.

Cf. A057655, A123689, A291259.

Let N(u,v,n) be the number of integer solutions (x,y) of (x-u)^2 + (y-v)^2 <= n. Then a(n) is the minimum of N(u,v,n) taken over 0 <= u <= 1/2 and 0 <= v <= u. Due to the symmetries of the square lattice one can limit the position (u,v) of the circle center within this triangle. The terms of the sequence were found by "brute force" search of the minimum of N(u,v,n) for (u,v) running through the triangular domain above.

A356462 a(n) is the maximum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

Original entry on oeis.org

1, 5, 9, 12, 14, 21, 21, 24, 28, 32, 37, 37, 41, 45, 48, 52, 52, 57, 61, 63, 69, 69, 72, 76, 78, 81, 89, 89, 92, 97, 97, 100, 104, 112, 112, 115, 116, 121, 122, 127, 129, 137, 137, 140, 144, 148, 148, 152, 155, 157, 161, 164, 169, 177, 177

Offset: 0

Bernard Montaron, Aug 08 2022

nonn

a(n) >= A057655(n).

The terms of square index of this sequence are such that a(n^2) = A123690(2n), e.g., a(9) = 32 = A123690(6).

			For n = 1 the maximum number of Z x Z lattice points inside the circle is a(1) = 5. The maximum is obtained with the circle centered at x = 0, y = 0.

Cf. A057655, A123690, A346993.

Let N(u,v,n) be the number of integer solutions (x,y) of (x-u)^2 + (y-v)^2 <= n. Then a(n) is the maximum of N(u,v,n) taken over 0 <= u <= 1/2 and 0 <= v <= u. The symmetries of the square lattice allow to limit the domain of the circle center (u,v) to this triangle. The terms of this sequence were found by "brute force" search of the maximum of N(u,v,n) for (u,v) in this triangular domain.

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A372511 Number of solutions to x^2 + y^2 <= n, where x, y are positive odd integers.

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A073092 Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.

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A156790 Number of first quadrant lattice squares inside the circle x^2+y^2=(2^n)^2.

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A175373 Partial sums of A175372.

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A256465 Number of points in a circle of squared radius n, points on the circle counted half.

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A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

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A363341 Number of positive integers k <= n such that round(n/k) is odd.

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A356435 a(n) is the minimum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

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A356462 a(n) is the maximum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

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