A358627 Triangle read by rows: T(n,k) is the number of edges 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.

9, 16, 40, 13, 20, 20, 19, 124, 17, 24, 64, 24, 140, 60, 204, 21, 28, 60, 28, 28, 74, 284, 39, 300, 25, 32, 32, 32, 176, 32, 292, 31, 68, 136, 436, 29, 36, 68, 36, 156, 84, 36, 53, 484, 158, 588, 67, 612, 33, 40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764, 37, 44, 44, 44, 140, 44, 316, 62, 44, 202, 740, 43, 884, 268, 148, 103, 980, 41

Offset: 1

Author

Scott R. Shannon, Nov 24 2022

Keywords

• nonn
• tabf

Comments

See A358556 for further details and images of the squares.

Examples

			The table begins:
9;
16, 40, 13;
20, 20, 19, 124, 17;
24, 64, 24, 140, 60, 204, 21;
28, 60, 28,  28, 74, 284, 39, 300,  25;
32, 32, 32, 176, 32, 292, 31,  68, 136, 436, 29;
36, 68, 36, 156, 84,  36, 53, 484, 158, 588, 67, 612,  33;
40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764,  37;
44, 44, 44, 140, 44, 316, 62,  44, 202, 740, 43, 884, 268, 148, 103, 980, 41;
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See the attached file for more examples.
		

Links

• Scott R. Shannon, Table for n=1..50.

Crossrefs

Cf. A358556 (regions), A358574 (vertices), A331452, A355798, A355838, A357058, A358407, A345459.

Formula

T(n,k) = A358574(n,k) + A358556(n,k) - 1 by Euler's formula.

T(n,2*n) = 4*(n + 1) + 1. The line cuts the square into two parts so one additional edge is created.

T(n,k) = 4*(n + 2) where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional edges are created.