A303644 -id:A303644 - cp's OEIS frontend

A303642 a(n) is the number of lattice points in Cartesian grid between circle of radius n and its inscribed square. The sides of the square are parallel to coordinate axes.

0, 0, 0, 20, 20, 28, 64, 72, 80, 80, 148, 148, 156, 248, 256, 264, 264, 380, 396, 404, 528, 552, 560, 700, 716, 740, 764, 928, 936, 960, 1148, 1180, 1196, 1212, 1440, 1448, 1472, 1700, 1740, 1764, 2000, 2040, 2064, 2104, 2380, 2396, 2428, 2720, 2760, 2784, 2832, 3156

Offset: 1

Kirill Ustyantsev, Apr 27 2018

nonn

If the sides of the inscribed square are parallel to bisector of coordinate axes we have a different sequence.

Kirill Ustyantsev, illustrated example

Cf. A302829, A303644, A303646.

import math
for n in range(1, 100):
 count = 0
 for x in range(0, n):
  for y in range(-n, n):
   if (x * x + y * y < n * n and x > n / math.sqrt(2)):
    count = count + 1
 print(4 * count)

Offset corrected by Andrey Zabolotskiy, May 05 2018

A302829 a(n) is the number of lattice points in a Cartesian grid between a circle of radius n and an inscribed square whose vertices lie on the coordinate axes.

Original entry on oeis.org

0, 0, 4, 8, 12, 28, 36, 52, 72, 88, 112, 128, 156, 192, 220, 252, 280, 324, 368, 408, 448, 504, 548, 592, 644, 708, 776, 828, 880, 952, 1016, 1096, 1164, 1236, 1324, 1388, 1472, 1548, 1648, 1736, 1808, 1912, 2004, 2116, 2212, 2300, 2408, 2508, 2624, 2728, 2860, 2976, 3076

Offset: 1

Kirill Ustyantsev, Apr 27 2018

nonn

Points are not lying on the borders of the circle and the square.

Note that if the square is rotated so that its sides are parallel to the coordinate axes, the resulting sequence is A303642 instead.

Kirill Ustyantsev, Desmos graph calculator

Cf. A034856, A281795, A303642, A303644, A303646.

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

for n in range (1, 100):
    count=0
    for x in range (0, n):
        for y in range (0, n):
            if (x*x+y*yn):
                count=count+1
    print(4*count)

a(n) = A281795(n) - 4*A034856(n). - Andrey Zabolotskiy, Apr 29 2018

A303646 a(n) is the number of lattice points in a Cartesian grid between a square with integer sides 2*n and its inscribed circle. The sides of the square are parallel to the bisector of coordinate axes.

Original entry on oeis.org

0, 0, 12, 12, 32, 32, 32, 68, 60, 104, 104, 104, 156, 148, 216, 216, 300, 292, 276, 368, 368, 468, 460, 452, 560, 544, 676, 668, 816, 792, 784, 932, 916, 1080, 1048, 1048, 1220, 1212, 1384, 1360, 1352, 1556, 1532, 1736, 1704, 1956, 1924, 1900, 2136, 2096, 2340, 2308

Offset: 1

Kirill Ustyantsev, Apr 27 2018

nonn

Rotating the square by 45 degrees (so that its vertices lie on the coordinate axes) results in sequence A303644 instead.

Kirill Ustyantsev, Graphic illustration

Cf. A302829, A303642, A303644.

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

import math
for n in range (1, 100):
   sqn = math.ceil(math.sqrt(2)*n)
   count = 0
   for x in range(-sqn, sqn):
       for y in range(-sqn, sqn):
           if (x*x+y*y>n*n and abs(x)+abs(y)

A303706 a(n) is the number of lattice points in a Cartesian grid between an equilateral triangle and an inscribed circle of radius n; one of the side of triangle is perpendicular to the X-axis; the circle's center is at the origin.

Original entry on oeis.org

0, 5, 14, 29, 42, 65, 94, 123, 154, 187, 234, 289, 328, 383, 436, 507, 572, 645, 716, 789, 884, 961, 1058, 1159, 1244, 1347, 1454, 1573, 1692, 1805, 1940, 2057, 2194, 2325, 2454, 2621, 2758, 2927, 3060, 3221, 3404, 3571, 3746, 3909, 4086, 4293, 4478, 4677, 4868, 5061, 5256, 5465, 5698, 5915

Offset: 1

Kirill Ustyantsev, Apr 29 2018

nonn

			For n = 2 we have 5 lattice points: (-1, 2); (-1, -2); (2, -1); (2, 1); (3, 0).

Kirill Ustyantsev, Geometric illustration

Cf. A303644, A303646.

a(n) = sum(x=-n+1, 2*n, sum(y=-2*n, 2*n, ((x^2+y^2) > n^2) && (3*y^2 < (x-2*n)^2))); \\ Michel Marcus, May 22 2018

import math
tan=math.sqrt(3)/3
for n in range (1,71):
  count=0
  for x in range (-n, 2*n):
   for y in range (-2*n, 2*n):
    if (x*x+y*y>n*n and y<-tan*x+2*tan*n and y>tan*x-2*tan*n and x>-n):
     count=count+1
  print(count)

cp's OEIS Frontend

A303642 a(n) is the number of lattice points in Cartesian grid between circle of radius n and its inscribed square. The sides of the square are parallel to coordinate axes.

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A302829 a(n) is the number of lattice points in a Cartesian grid between a circle of radius n and an inscribed square whose vertices lie on the coordinate axes.

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A303646 a(n) is the number of lattice points in a Cartesian grid between a square with integer sides 2*n and its inscribed circle. The sides of the square are parallel to the bisector of coordinate axes.

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A303706 a(n) is the number of lattice points in a Cartesian grid between an equilateral triangle and an inscribed circle of radius n; one of the side of triangle is perpendicular to the X-axis; the circle's center is at the origin.

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PARI

Python