A119439 -id:A119439 - cp's OEIS frontend

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

6, 12, 8, 48, 4, 10, 108, 16, 4, 12, 248, 36, 4, 4, 14, 428, 64, 20, 4, 4, 16, 764, 100, 44, 4, 4, 4, 18, 1196, 204, 36, 24, 4, 4, 4, 20, 1900, 252, 64, 52, 4, 4, 4, 4, 22, 2668, 396, 124, 40, 28, 4, 4, 4, 4, 24, 3824, 572, 200, 20, 60, 4, 4, 4, 4, 4, 26, 5244, 780, 236, 76, 44, 32

Offset: 2

Franklin T. Adams-Watters, May 19 2006

			From _Seiichi Manyama_, Nov 26 2017: (Start)
The table starts:
  n\k|   2    3   4   5   6   7   8
  ---+------------------------------
   2 |   6;
   3 |  12,   8;
   4 |  48,   4, 10;
   5 | 108   16,  4, 12;
   6 | 248,  36,  4,  4, 14;
   7 | 428,  64, 20,  4,  4, 16;
   8 | 764, 100, 44,  4,  4,  4, 18; (End)

Seiichi Manyama, Rows n = 2..141, flattened
S. Mustonen, On lines and their intersection points in a rectangular grid of points [From _Seppo Mustonen_, Apr 18 2009]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Row sums A018808; columns A018809-A018817. See A119439 for another version.

T(n,k) = 1/2 (f(n, k+1) - 2 f(n, k) + f(n, k-1)) where f(n, k) = Sum_{-n < kx < n, -n < ky < n, gcd(x, y)=1} (n - |kx|)*(n - |ky|). [Seppo Mustonen, Apr 18 2009]

An incorrect formula removed by Seppo Mustonen, Apr 25 2009

A119438 Number of sets of points determined by the intersection of a line with an n X n grid of points.

Original entry on oeis.org

1, 2, 11, 30, 79, 166, 343, 586, 1003, 1574, 2407, 3418, 4867, 6630, 9027, 11794, 15203, 19190, 24251, 29906, 36911, 44830, 54071, 64178, 76251, 89574, 105051, 121762, 140751, 161478, 185367, 210906, 240075, 271678, 306635, 343706, 384667

Offset: 0

Franklin T. Adams-Watters, May 19 2006

nonn

Each singleton point is determined by all but finitely many of the family of lines passing through that point and the empty set is determined by any randomly positioned line.

Cf. A018808, A119439.

a(n) = A018808(n) + n^2 + 1.

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

cp's OEIS Frontend

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

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A119438 Number of sets of points determined by the intersection of a line with an n X n grid of points.

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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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