A360352 -id:A360352 - 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.

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

A360353 Number of edges 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

16, 244, 2580, 15788, 67144, 227888, 668008, 1712960, 3968028

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.

Cf. A360351 (vertices), A360352 (regions), A360354 (k-gons), A360350 (distinct circles), A359934.

a(n) = A360351(n) + A360352(n) - 1 by Euler's formula.

A372615 Number of regions 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

7, 121, 624, 2368, 7912, 17115, 38461, 63184, 115614, 170149, 285715, 371982, 588043

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.

The vertices of the initial equilateral triangle are indicated by small circles in the illustrations here.

Scott R. Shannon, Image for n = 0.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A372614 (vertices), A372616 (edges), A372617 (k-gons), A372682 (number of circles), A371374, A353782, A360352.

a(n) = A372616(n) - A372614(n) + 1 by Euler's formula.

A372978 Number of regions 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

45, 628, 4633, 15476, 41561, 98808, 206317, 333272, 590181

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.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn as white dots for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.

Cf. A372977 (vertices), A372979 (edges), A372980 (k-gons), A372981 (circles), A372615, A371374, A353782, A360352.

a(n) = A372979(n) - A372977(n) + 1 by Euler's formula.

A372732 Number of regions 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

7, 88, 438, 1567, 3910, 9504, 17905, 32833, 54066, 86365, 128380, 190152, 262699, 365761, 486711, 642154

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.

Scott R. Shannon, Image for n = 0.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A372731 (vertices), A372733 (edges), A372734 (k-gons), A372735 (number of circles), A372615, A371374, A353782, A360352.

a(n) = A372733(n) - A372731(n) + 1 by Euler's formula.

A373107 Number of regions 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

12, 136, 932, 3252, 9144, 20032, 41260, 71832, 121460, 188656, 291488

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.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn as white dots for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 7.

Cf. A373106 (vertices), A373108 (edges), A373109 (k-gons), A373110 (circles), A372978, A372732, A358782, A362234, A360352.

a(n) = A373108(n) - A373106(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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A360353 Number of edges 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.

Views

Author

Keywords

Comments

Crossrefs

Formula

A372615 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A372978 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A373107 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A373107 Number of regions 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.