cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A386559 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of points where lines intersect in the resulting graph.

Original entry on oeis.org

5, 65, 381, 1213, 3033, 6105, 12285, 20789, 33705, 51065, 79797, 110817, 161549, 216985, 284269, 367925, 489953, 609225, 785045, 952877, 1157749, 1404473
Offset: 1

Views

Author

Scott R. Shannon, Jul 26 2025

Keywords

Comments

It appears that for n >= 3 the intersection that is furthest from the origin is formed by the crossing of the lines y = n/(n-1)*x + n and y = (n-1)/(n-2)*x - (n-1), along with the seven other symmetrically equivalent intersections. These intersections have a distance from the origin of approximately sqrt(8)*n^3 as n -> infinity.

Links

Crossrefs

Cf. A386560 (regions), A386561 (edges), A386562 (k-gons), A146212, A347750, A344657, A345649.

Formula

a(n) = A386561(n) - A386560(n) + 1 by Euler's formula.

A386560 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of finite regions created in the resulting graph.

Original entry on oeis.org

4, 72, 424, 1396, 3536, 7292, 14272, 24332, 39356, 59920, 91348, 128084, 182664, 245804, 323116, 418552, 547820, 684680, 869388, 1060892, 1289564, 1560920
Offset: 1

Views

Author

Scott R. Shannon, Jul 26 2025

Keywords

Links

Crossrefs

Cf. A386559 (vertices), A386561 (edges), A386562 (k-gons), A344993, A344857, A344279, A345459.

Formula

a(n) = A386561(n) - A386559(n) + 1 by Euler's formula.

A386562 Irregular table read by rows: Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: T(n,k) is the number of k-sided finite polygons formed, for k>=3, in the resulting graph.

Original entry on oeis.org

4, 44, 24, 4, 184, 216, 24, 560, 780, 56, 1456, 1844, 224, 12, 3100, 3788, 376, 24, 4, 5860, 7100, 1148, 156, 8, 9860, 12436, 1848, 164, 20, 4, 16044, 19732, 3100, 460, 16, 4, 24744, 29568, 5048, 516, 32, 12, 36780, 43472, 9608, 1400, 68, 20, 52296, 61244, 12628, 1784, 116, 16
Offset: 1

Views

Author

Scott R. Shannon, Jul 26 2025

Keywords

Comments

For graphs up to n = 22 the k-gons with the largest number of sides are 12-gons, first appearing for n = 20. The behavior of this maximum value as n -> infinity is unknown.
See A386559 and A386560 for other images of the graphs.

Examples

			The table begins:
4;
44, 24, 4;
184, 216, 24;
560, 780, 56;
1456, 1844, 224, 12;
3100, 3788, 376, 24, 4;
5860, 7100, 1148, 156, 8;
9860, 12436, 1848, 164, 20, 4;
16044, 19732, 3100, 460, 16, 4;
24744, 29568, 5048, 516, 32, 12;
36780, 43472, 9608, 1400, 68, 20;
52296, 61244, 12628, 1784, 116, 16;
72492, 85672, 20424, 3792, 268, 16;
97812, 115000, 27796, 4820, 344, 24, 8;
129416, 151184, 35716, 6240, 532, 28;
167712, 195816, 46380, 7956, 644, 44;
.
.

Links

  • Scott R. Shannon, Image for n = 2. The integer coordinates are highlighted as white dots while the outer open regions, which are not counted, are darkened.

Crossrefs

Cf. A386559 (vertices), A386560 (regions), A386561 (edges), A344938, A346446.

Formula

Sum of row n = A386560(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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