A383462 Triangle read by rows: T(n,k) (n >= 3, 2 <= k <= n-1) = number of vertices where k lines cross in the planar graph formed when every pair of vertices of a regular n-gon are joined by an infinite line.
3, 1, 4, 10, 0, 5, 30, 1, 0, 6, 84, 0, 0, 0, 7, 120, 16, 1, 0, 0, 8, 324, 0, 0, 0, 0, 0, 9, 420, 40, 0, 1, 0, 0, 0, 10, 880, 0, 0, 0, 0, 0, 0, 0, 11, 708, 156, 24, 0, 1, 0, 0, 0, 0, 12, 1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14
Offset: 3
Examples
Triangle begins:
3;
1, 4;
10, 0, 5;
30, 1, 0, 6;
84, 0, 0, 0, 7;
120, 16, 1, 0, 0, 8;
324, 0, 0, 0, 0, 0, 9;
420, 40, 0, 1, 0, 0, 0, 10;
880, 0, 0, 0, 0, 0, 0, 0, 11;
708, 156, 24, 0, 1, 0, 0, 0, 0, 12;
1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14;
3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15;
3408, 544, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 16;
6664, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
4572, 756, 108, 108, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 18;
10944, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
9840, 1280, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 20;
.
.
See the attached table for rows 3 to 100.
For n = 8, we may classify the vertices by degree and according to whether they are outside, on, or inside the octagon:
V2 V3 V4 V5 V6 V7
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outside 80 8
on 0 0 0 0 0 8
inside 40 8 1 0 0 0
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totals 120 16 1 0 0 8
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Grand total: 145 = A146212(8)
In general, for n >= 3, the counts for inside the defining polygon are given by row n of A292105, the total number on or inside the polygon by A007569, and the number outside by A146213.
Links
- Scott R. Shannon, Table of n, a(n) for n = 3..4853
- Scott R. Shannon, Formatted table for rows 3 to 100.
- Scott R. Shannon, Image of the 5-gon.
- Scott R. Shannon, Image of the 6-gon.
- Scott R. Shannon, Image of the 7-gon.
- Scott R. Shannon, Image of the 8-gon.
Comments