login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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.
3

%I #25 Nov 24 2022 12:51:06

%S 8,12,20,12,16,16,16,64,16,20,36,20,68,36,100,20,24,36,24,24,44,144,

%T 29,144,24,28,28,28,92,28,140,28,44,76,208,28,32,44,32,84,52,32,39,

%U 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

%N 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.

%C See A358556 for further details.

%H Scott R. Shannon, <a href="/A358574/a358574.txt">Table for n=1..50</a>.

%H Scott R. Shannon, <a href="/A358574/a358574.png">Image for T(2,3) = 20</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_1.png">Image for T(4,6) = 36</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_2.png">Image for T(7,9) = 240</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_3.png">Image for T(10,19) = 584</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_4.png">Image for T(11,20) = 90</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_5.png">Image for T(20,11) = 308</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_6.png">Image for T(20,30) = 100</a>.

%H Scott R. Shannon, <a href="/A358574/a358574_7.png">Image for T(20,31) = 2220</a>.

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

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

%F 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.

%e The table begins:

%e 8;

%e 12, 20, 12;

%e 16, 16, 16, 64, 16;

%e 20, 36, 20, 68, 36, 100, 20;

%e 24, 36, 24, 24, 44, 144, 29, 144, 24;

%e 28, 28, 28, 92, 28, 140, 28, 44, 76, 208, 28;

%e 32, 44, 32, 84, 52, 32, 39, 240, 88, 292, 46, 296, 32;

%e 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;

%e 40, 40, 40, 80, 40, 164, 47, 40, 112, 364, 40, 436, 144, 88, 67, 472, 40;

%e .

%e .

%e See the attached file for more examples.

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

%K nonn,tabf

%O 1,1

%A _Scott R. Shannon_, Nov 23 2022