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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.

%I A358574 #25 Nov 24 2022 12:51:06
%S A358574 8,12,20,12,16,16,16,64,16,20,36,20,68,36,100,20,24,36,24,24,44,144,
%T A358574 29,144,24,28,28,28,92,28,140,28,44,76,208,28,32,44,32,84,52,32,39,
%U A358574 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 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.
%C A358574 See A358556 for further details.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574.txt">Table for n=1..50</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574.png">Image for T(2,3) = 20</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_1.png">Image for T(4,6) = 36</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_2.png">Image for T(7,9) = 240</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_3.png">Image for T(10,19) = 584</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_4.png">Image for T(11,20) = 90</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_5.png">Image for T(20,11) = 308</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_6.png">Image for T(20,30) = 100</a>.
%H A358574 Scott R. Shannon, <a href="/A358574/a358574_7.png">Image for T(20,31) = 2220</a>.
%F A358574 T(n,k) = A358627(n,k) - A358556(n,k) + 1 by Euler's formula.
%F A358574 T(n,2*n) = 4*(n + 1). The line cuts the square into two parts so no new vertices are created.
%F A358574 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 A358574 The table begins:
%e A358574 8;
%e A358574 12, 20, 12;
%e A358574 16, 16, 16, 64, 16;
%e A358574 20, 36, 20, 68, 36, 100, 20;
%e A358574 24, 36, 24, 24, 44, 144, 29, 144, 24;
%e A358574 28, 28, 28, 92, 28, 140, 28,  44,  76, 208, 28;
%e A358574 32, 44, 32, 84, 52,  32, 39, 240,  88, 292, 46, 296,  32;
%e A358574 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;
%e A358574 40, 40, 40, 80, 40, 164, 47,  40, 112, 364, 40, 436, 144,  88, 67, 472, 40;
%e A358574 .
%e A358574 .
%e A358574 See the attached file for more examples.
%Y A358574 Cf. A358556 (regions), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.
%K A358574 nonn,tabf
%O A358574 1,1
%A A358574 _Scott R. Shannon_, Nov 23 2022