A360353 -id:A360353 - cp's OEIS frontend

A360351 Number of vertices among all distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

5, 77, 1045, 6885, 30265, 104421, 309973, 800185, 1862053

Offset: 2

Scott R. Shannon, Feb 03 2023

A circle is constructed for every pair of points on the n X n grid, the points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the n X n grid is A360350(n).

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A360352 (regions), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359932.

a(n) = A360353(n) - A360352(n) + 1 by Euler's formula.

A360352 Number of regions among all distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

Original entry on oeis.org

12, 168, 1536, 8904, 36880, 123468, 358036, 912776, 2105976

Offset: 2

Scott R. Shannon, Feb 04 2023

A circle is constructed for every pair of points on the n X n grid, the points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the n X n grid is A360350(n).

Scott R. Shannon, Image for n = 2. In this and other images the n X n grid points are shown as white dots.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A360351 (vertices), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359933.

a(n) = A360353(n) - A360351(n) + 1 by Euler's formula.

A360354 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from an n x n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

Original entry on oeis.org

8, 4, 40, 108, 20, 92, 904, 456, 76, 0, 8, 200, 4540, 3400, 652, 100, 4, 8, 404, 17244, 15324, 3148, 628, 116, 16, 528, 54252, 54476, 11672, 2152, 332, 44, 12, 972, 151992, 158468, 37244, 7940, 1120, 224, 48, 12, 16, 1404, 379488, 404148, 103436, 20216, 3316, 600, 132, 20, 16

Offset: 2

Scott R. Shannon, Feb 04 2023

A circle is constructed for every pair of points on the n X n grid, the points lying at the ends of a diameter of the circle.
The number of distinct circles constructed from the n x n grid is A360350(n).
See A360351 and A360352 for images of the circles.

			The table begins:
8, 4;
40, 108, 20;
92, 904, 456, 76, 0, 8;
200, 4540, 3400, 652, 100, 4, 8;
404, 17244, 15324, 3148, 628, 116, 16;
528, 54252, 54476, 11672, 2152, 332, 44, 12;
972, 151992, 158468, 37244, 7940, 1120, 224, 48, 12, 16;
1404, 379488, 404148, 103436, 20216, 3316, 600, 132, 20, 16;
1896, 868460, 923656, 252096, 49848, 7916, 1744, 276, 84;
.
.

Cf. A360351 (vertices), A360352 (regions), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359935.

Sum of row n = A360352(n).

A360350 Number of distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

Original entry on oeis.org

5, 26, 79, 185, 366, 653, 1077, 1678, 2494, 3571, 4959, 6718, 8889, 11541, 14740, 18553, 23027, 28278, 34351, 41352, 49356, 58454, 68732, 80330, 93304, 107757, 123815, 141605, 161211, 182795, 206393, 232190, 260331, 290907, 324090, 360080, 398856, 440655, 485655

Offset: 2

Scott R. Shannon, Feb 03 2023

nonn

A circle is constructed for every pair of points on the n X n grid, the points lying at the ends of a diameter of the circle.
No formula for a(n) is known.
See A360351 and A360352 for images of the resulting vertices and regions.

N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A360351 (vertices), A360352 (regions), A360353 (edges), A360354 (k-gons), A359931.

a(n) = { my (p = vector(n^2, k, (k-1)%n + ((k-1)\n)*I)); #setbinop((i,j)->[i+j, norm(i-j)], p)-n^2; } \\ Rémy Sigrist, Sep 24 2023

More terms from Rémy Sigrist, Sep 24 2023

A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

16, 196, 1608, 5784, 16848, 37300, 78420, 136920, 233336, 363200, 565700

Offset: 0

Scott R. Shannon, May 25 2024

A circle is constructed for every pair of the 4 + 4*n points, the two points lying at the ends of a diameter of the circle.

See A373106 and A373107 for images of the circles.

Cf. A373106 (vertices), A373107 (regions), A373109 (k-gons), A373110 (circles), A372979, A372733, A358783, A362235, A360353.

a(n) = A373106(n) + A373107(n) - 1 by Euler's formula.

A372616 Number of curved edges among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

Original entry on oeis.org

12, 207, 1104, 4359, 14880, 32523, 73662, 121605, 223290, 329286, 554286, 722841, 1145091

Offset: 0

Scott R. Shannon, May 07 2024

A circle is constructed for every pair of the 3 + 3*n points, the first point defines the circle's center while the second the radius distance.

See A372614 and A372615 for images of the circles.

Cf. A372614 (vertices), A372615 (regions), A372617 (k-gons), A372682 (number of circles), A371375, A356358, A360353.

a(n) = A372614(n) + A372615(n) - 1 by Euler's formula.

A372979 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

Original entry on oeis.org

84, 1180, 8836, 29980, 80564, 193172, 403780, 654196, 1159780

Offset: 0

Scott R. Shannon, May 19 2024

A circle is constructed for every pair of the 4 + 4*n points, the first point defines the circle's center while the second the radius distance.

See A372977 and A372978 for images of the circles.

Cf. A372977 (vertices), A372978 (regions), A372980 (k-gons), A372981 (circles), A372616, A371375, A356358, A360353.

a(n) = A372977(n) + A372978(n) - 1 by Euler's formula.

A372733 Number of curved edges among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

12, 138, 738, 2838, 7194, 17904, 33954, 62868, 103866, 167280, 248826, 370458, 511806, 715905, 952608, 1260366

Offset: 0

Scott R. Shannon, May 12 2024

A circle is constructed for every pair of the 3 + 3*n points, the two points lying at the ends of a diameter of the circle.

See A372731 and A372732 for images of the circles.

Cf. A372731 (vertices), A372732 (regions), A372734 (k-gons), A372735 (number of circles), A372616, A371375, A356358, A360353.

a(n) = A372731(n) + A372732(n) - 1 by Euler's formula.

cp's OEIS Frontend

A360351 Number of vertices among all distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

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A360352 Number of regions among all distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

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A360354 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from an n x n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

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A360350 Number of distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

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A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square when every pair of the 4 + 4*n points are connected by a circle and where the points lie at the ends of the circle's diameter.

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A372616 Number of curved edges among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

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A372979 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

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A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.