A358627 - OEIS

%I #16 Nov 24 2022 12:50:55

%S 9,16,40,13,20,20,19,124,17,24,64,24,140,60,204,21,28,60,28,28,74,284,

%T 39,300,25,32,32,32,176,32,292,31,68,136,436,29,36,68,36,156,84,36,53,

%U 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

%N 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.

%C See A358556 for further details and images of the squares.

%H Scott R. Shannon, <a href="/A358627/a358627.txt">Table for n=1..50</a>.

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

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

%F 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.

%e The table begins:

%e 9;

%e 16, 40, 13;

%e 20, 20, 19, 124, 17;

%e 24, 64, 24, 140, 60, 204, 21;

%e 28, 60, 28,  28, 74, 284, 39, 300,  25;

%e 32, 32, 32, 176, 32, 292, 31,  68, 136, 436, 29;

%e 36, 68, 36, 156, 84,  36, 53, 484, 158, 588, 67, 612,  33;

%e 40, 72, 40, 144, 80, 328, 40, 520, 180, 648, 76, 752, 232, 764,  37;

%e 44, 44, 44, 140, 44, 316, 62,  44, 202, 740, 43, 884, 268, 148, 103, 980, 41;

%e .

%e .

%e See the attached file for more examples.

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

%K nonn,tabf

%O 1,1

%A _Scott R. Shannon_, Nov 24 2022