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%I #30 Nov 24 2022 12:50:45
%S 2,5,21,2,5,5,4,61,2,5,29,5,73,25,105,2,5,25,5,5,31,141,11,157,2,5,5,
%T 5,85,5,153,4,25,61,229,2,5,25,5,73,33,5,15,245,71,297,22,317,2,5,25,
%U 5,65,29,165,5,269,81,333,25,385,109,401,2,5,5,5,61,5,153,16,5,91,377,4,449,125,61,37,509,2
%N Triangle read by rows: T(n,k) is the number of regions formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.
%C The starting point can be any of the 4*n points around the square as changing the starting point simply rotates and/or reflects the resulting pattern formed by the path to one of the four orthogonal directions around the square; this does not change the number of regions formed by the path.
%C The number of times the path formed by the line touches and leaves the edges of the square is lcm(4*n,k)/k. For k >= n this is the number of points in the star-shaped pattern formed by the path.
%C The table starts with k = 2 as T(n,1) = 5 for all values of n. The maximum k is 2*n as T(n,2*n + m) = T(n,2*n - m).
%H Scott R. Shannon, <a href="/A358556/a358556.txt">Table for n=1..50</a>.
%H Scott R. Shannon, <a href="/A358556/a358556.jpg">Image for T(2,3) = 21</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_1.jpg">Image for T(4,6) = 25</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_2.jpg">Image for T(7,9) = 245</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_3.jpg">Image for T(10,19) = 629</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_4.jpg">Image for T(11,20) = 55</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_5.jpg">Image for T(20,11) = 269</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_6.jpg">Image for T(20,30) = 25</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_7.jpg">Image for T(20,31) = 2277</a>.
%H Scott R. Shannon, <a href="/A358556/a358556_8.jpg">Image for T(50,61) = 11933</a>.
%F T(n,k) = A358627(n,k) - A358574(n,k) + 1 by Euler's formula.
%F T(n,2*n) = 2. The line cuts the square into two parts.
%F T(n,k) = 5 where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional triangles are created.
%e The table begins:
%e 2;
%e 5, 21, 2;
%e 5, 5 4, 61, 2;
%e 5, 29, 5, 73, 25, 105, 2;
%e 5, 25, 5, 5, 31, 141, 11, 157, 2;
%e 5, 5, 5, 85, 5, 153, 4, 25, 61, 229, 2;
%e 5, 25, 5, 73, 33, 5, 15, 245, 71, 297, 22, 317, 2;
%e 5, 25, 5, 65, 29, 165, 5, 269, 81, 333, 25, 385, 109, 401, 2;
%e 5, 5, 5, 61, 5, 153, 16, 5, 91, 377, 4, 449, 125, 61, 37, 509, 2;
%e 5, 25, 5, 5, 25, 137, 5, 285, 5, 385, 31, 501, 141, 25, 11, 613, 169, 629, 2;
%e .
%e .
%e See the attached file for more examples.
%Y Cf. A358574 (vertices), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.
%K nonn,tabf
%O 1,1
%A _Scott R. Shannon_, Nov 22 2022
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