A358574 Triangle read by rows: T(n,k) is the number of vertices 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.

8, 12, 20, 12, 16, 16, 16, 64, 16, 20, 36, 20, 68, 36, 100, 20, 24, 36, 24, 24, 44, 144, 29, 144, 24, 28, 28, 28, 92, 28, 140, 28, 44, 76, 208, 28, 32, 44, 32, 84, 52, 32, 39, 240, 88, 292, 46, 296, 32, 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36, 40, 40, 40, 80, 40, 164, 47, 40, 112, 364, 40, 436, 144, 88, 67, 472, 40

Offset: 1

Author

Scott R. Shannon, Nov 23 2022

Keywords

• nonn
• tabf

Comments

See A358556 for further details.

Examples

			The table begins:
8;
12, 20, 12;
16, 16, 16, 64, 16;
20, 36, 20, 68, 36, 100, 20;
24, 36, 24, 24, 44, 144, 29, 144, 24;
28, 28, 28, 92, 28, 140, 28,  44,  76, 208, 28;
32, 44, 32, 84, 52,  32, 39, 240,  88, 292, 46, 296,  32;
36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;
40, 40, 40, 80, 40, 164, 47,  40, 112, 364, 40, 436, 144,  88, 67, 472, 40;
.
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See the attached file for more examples.
		

Links

• Scott R. Shannon, Table for n=1..50.
• Scott R. Shannon, Image for T(2,3) = 20.
• Scott R. Shannon, Image for T(4,6) = 36.
• Scott R. Shannon, Image for T(7,9) = 240.
• Scott R. Shannon, Image for T(10,19) = 584.
• Scott R. Shannon, Image for T(11,20) = 90.
• Scott R. Shannon, Image for T(20,11) = 308.
• Scott R. Shannon, Image for T(20,30) = 100.
• Scott R. Shannon, Image for T(20,31) = 2220.

Crossrefs

Cf. A358556 (regions), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

Formula

T(n,k) = A358627(n,k) - A358556(n,k) + 1 by Euler's formula.

T(n,2*n) = 4*(n + 1). The line cuts the square into two parts so no new vertices are created.

T(n,k) = 4*(n + 1) where k <= n and k|(4*n). Four lines cut across the square's corners so no new vertices are created.