A352533 - cp's OEIS frontend

A352533 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex is fully connected to all other vertices in counterclockwise order before the next vertex, in counterclockwise order, is chosen.

%I A352533 #28 Mar 20 2022 11:18:00
%S A352533 1,1,2,4,1,2,3,5,8,11,1,2,3,4,6,9,13,16,20,24,1,2,3,4,5,7,10,14,19,22,
%T A352533 27,34,38,45,50,1,2,3,4,5,6,8,11,15,20,26,29,34,39,46,50,56,62,67,74,
%U A352533 80,1,2,3,4,5,6,7,9,12,16,21,27,34,37,42,49,58,69,73,80,90,103,108,117,130,136,147,154
%N A352533 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex is fully connected to all other vertices in counterclockwise order before the next vertex, in counterclockwise order, is chosen.
%C A352533 To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to all other vertices where those vertices are chosen in a counterclockwise direction around the n-gon from the starting vertex. Once all those diagonals are drawn the next neighboring vertex, in a counterclockwise direction from the starting vertex, is chosen. This vertex is then connected to all other vertices in the same way. This method is repeated until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533_1.txt">Table for n=3..100</a>.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533_4.png">Image for the 7-gon after first vertex is connected to all other vertices</a>. Total regions = 5.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533.png">Image for the 7-gon after second vertex is connected to all other vertices</a>. Total regions = 19.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533_1.png">Image for the 7-gon after third vertex is connected to all other vertices</a>. Total regions = 34.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533_2.png">Image for the 7-gon after fourth vertex is connected to all other vertices</a>. Total regions = 45.
%H A352533 Scott R. Shannon, <a href="/A352533/a352533_3.png">Image for the 7-gon after fifth vertex is connected to all other vertices</a>. Total regions = 50.
%F A352533 The last term in each row n = A007678(n).
%e A352533 The table begins:
%e A352533 1;
%e A352533 1,2,4;
%e A352533 1,2,3,5,8,11;
%e A352533 1,2,3,4,6,9,13,16,20,24;
%e A352533 1,2,3,4,5,7,10,14,19,22,27,34,38,45,50;
%e A352533 1,2,3,4,5,6,8,11,15,20,26,29,34,39,46,50,56,62,67,74,80;
%e A352533 1,2,3,4,5,6,7,9,12,16,21,27,34,37,42,49,58,69,73,80,90,103,108,117,130,136,147,154;
%e A352533 1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,46,51,58,65,75,86,90,97,107,116,130,135,143, \
%e A352533       154,164,170,180,194,201,212,220;
%e A352533 1,2,3,4,5,6,7,8,9,11,14,18,23,29,36,44,53,56,61,68,77,88,101,116,120,127,137,150, \
%e A352533       166,185,190,199,212,229,250,256,267,283,304,311,324,343,351,366,375;
%e A352533 .
%e A352533 .
%e A352533 See the linked file for the table up to n=100. See the linked images for examples of the 7-gon.
%Y A352533 Cf. A007678, A350000, A344857.
%K A352533 nonn,tabf
%O A352533 3,3
%A A352533 _Scott R. Shannon_, Mar 19 2022

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