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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.

%I A383462 #27 Jun 21 2025 14:15:42
%S A383462 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,
%T A383462 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,
%U A383462 0,0,0,0,0,0,0,0,0,13,1890,280,0,0,0,1,0,0,0,0,0,14
%N 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.
%C A383462 For other illustrations see A146212, A344857, A292105.
%H A383462 Scott R. Shannon, <a href="/A383462/b383462.txt">Table of n, a(n) for n = 3..4853</a>
%H A383462 Scott R. Shannon, <a href="/A383462/a383462_1.txt">Formatted table for rows 3 to 100</a>.
%H A383462 Scott R. Shannon, <a href="/A383462/a383462.png">Image of the 5-gon</a>.
%H A383462 Scott R. Shannon, <a href="/A383462/a383462_1.png">Image of the 6-gon</a>.
%H A383462 Scott R. Shannon, <a href="/A383462/a383462_2.png">Image of the 7-gon</a>.
%H A383462 Scott R. Shannon, <a href="/A383462/a383462_3.png">Image of the 8-gon</a>.
%e A383462 Triangle begins:
%e A383462    3;
%e A383462    1, 4;
%e A383462    10, 0, 5;
%e A383462    30, 1, 0, 6;
%e A383462    84, 0, 0, 0, 7;
%e A383462    120, 16, 1, 0, 0, 8;
%e A383462    324, 0, 0, 0, 0, 0, 9;
%e A383462    420, 40, 0, 1, 0, 0, 0, 10;
%e A383462    880, 0, 0, 0, 0, 0, 0, 0, 11;
%e A383462    708, 156, 24, 0, 1, 0, 0, 0, 0, 12;
%e A383462    1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
%e A383462    1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14;
%e A383462    3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15;
%e A383462    3408, 544, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 16;
%e A383462    6664, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
%e A383462    4572, 756, 108, 108, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 18;
%e A383462    10944, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
%e A383462    9840, 1280, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 20;
%e A383462    .
%e A383462    .
%e A383462 See the attached table for rows 3 to 100.
%e A383462 For n = 8, we may classify the vertices by degree and according to whether they are outside, on, or inside the octagon:
%e A383462                 V2      V3      V4      V5      V6      V7
%e A383462 ----------------------------------------------------------
%e A383462    outside      80      8
%e A383462    on           0       0       0       0       0       8
%e A383462    inside       40      8       1       0       0       0
%e A383462 ----------------------------------------------------------
%e A383462    totals       120     16      1       0       0       8
%e A383462 ----------------------------------------------------------
%e A383462    Grand total: 145 = A146212(8)
%e A383462 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.
%Y A383462 Row sums are A146212.
%Y A383462 Cf. A007569, A146213, A292105, A344857.
%K A383462 nonn,tabl
%O A383462 3,1
%A A383462 _Scott R. Shannon_ and _N. J. A. Sloane_, Jun 07 2025