A292104 -id:A292104 - cp's OEIS frontend

A371264 Irregular triangle read by rows: T(n,k) is the number of internal vertices in the graph A371254(n) that are created by the crossing of k arcs, with k>=2.

0, 0, 0, 1, 0, 5, 5, 0, 0, 0, 0, 1, 49, 14, 48, 8, 171, 27, 0, 0, 0, 0, 0, 1, 190, 20, 484, 55, 360, 12, 0, 0, 12, 0, 0, 0, 0, 0, 1, 1027, 91, 1078, 70, 1830, 120, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2000, 112, 3052, 204, 3114, 90, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Scott R. Shannon, Mar 18 2024

See A371254 for images of the graphs.

			The table begins:
0;
0;
0, 1;
0;
5, 5;
0, 0, 0, 0, 1;
49, 14;
48, 8;
171, 27, 0, 0, 0, 0, 0, 1;
190, 20;
484, 55;
360, 12, 0, 0, 12, 0, 0, 0, 0, 0, 1;
1027, 91;
1078, 70;
1830, 120, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0 1;
2000, 112;
3052, 204;
3114, 90, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
5662, 285;
5740, 240;
8610, 378, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
8888, 330;
12995, 506;
12312, 336, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18650, 650;
18668, 572;
25596, 810, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                        \\ 0,  1;
25928, 728;
34887, 1015;
32580, 510, 0, 0, 150, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                             \\ 0, 0, 0, 0, 0, 1;
46097, 1240;
46464, 1120;
.
.

Cf. A371254, A292105, A335102, A292104.

Sum of row(n) = A371254(n) - n;

A352434 The number of simple vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 6, 8, 8, 10, 8, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 20, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 32, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 44, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 56, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 68, 72, 72, 74, 74, 76

Offset: 1

Scott R. Shannon, Mar 16 2022

nonn

Excluding a(2), which has its simple vertex at the center of the 4-gon, the terms predominantly follow a pattern of pairs of two equal numbers and where the pair values increment by two. The second term of each pair corresponds to 2n-gons where n is a multiple of 2. These 2n-gons have two vertices that are on the same horizontal line as the central non-simple vertex thus the line joining them will not form a new simple vertex with the central vertical diagonal. Therefore in general a(2*k) = a(2*k-1), k>=1. However this rule is broken when n is a multiple of 12 - for these 2n-gons two of the horizontal lines connecting the left-side and right-side vertices also intersect two non-central diagonals and thus two simple vertices are removed. See the linked image of the 24-gon.

			a(2) = 1 as the 4-gon (square) has one simple vertex at its center when all its vertices are connected by lines.
a(3) = 2 as the 6-gon (hexagon) has two simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.
a(7) = 6 as the 14-gon has six simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.

Scott R. Shannon, Image of the 6-gon.
Scott R. Shannon, Image of the 10-gon.
Scott R. Shannon, Image of the 14-gon.
Scott R. Shannon, Image of the 24-gon.

Cf. A351924 (all vertices on diagonal), A352144 (all simple vertices), A292104, A007569, A006561, A146212.

A292103 Number of points that are the intersections of exactly two semicircles in the configuration A290447(n).

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 123, 195, 285, 420, 586, 818, 1110, 1451, 1846, 2361, 2956, 3704, 4567, 5530, 6631, 7963, 9443, 11113, 13005, 15111, 17450, 20167, 23064, 26396, 30053, 34046, 38447, 43230, 48245, 53890, 60061, 66703, 73713, 81503, 89746

Offset: 1

N. J. A. Sloane, Sep 14 2017

nonn

No formula or recurrence is known.

Needs a b-file (A290867 gives first 100 terms).

Cf. A290447, A292104.

Column k=2 of triangle in A290867.

cp's OEIS Frontend

A371264 Irregular triangle read by rows: T(n,k) is the number of internal vertices in the graph A371254(n) that are created by the crossing of k arcs, with k>=2.

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A352434 The number of simple vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

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A292103 Number of points that are the intersections of exactly two semicircles in the configuration A290447(n).

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