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%I #21 Jan 06 2024 12:02:13
%S 3,6,4,22,9,5,108,69,15,6,300,345,215,25,7,919,1337,1285,397,49,8,
%T 1626,2885,4435,2461,1008,65,9,3558,7445,11310,7873,5817,1601,144,10,
%U 5824,12833,24490,21271,19677,9225,3069,181,11,9843,23365,46610,46183,49973,33201,17316,4401,352,12
%N Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of these n*k points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of vertices in the resulting planar graph.
%C See A366483 and A367276 for other images of the n-gons.
%H Scott R. Shannon, <a href="/A367302/a367302.png">Image for T(5,5)</a>.
%H Scott R. Shannon, <a href="/A367302/a367302_1.png">Image for T(8,3)</a>.
%H Scott R. Shannon, <a href="/A367302/a367302_2.png">Image for T(12,2)</a>.
%F a(n,k) = A367305(n,k) - A367304(n,k) + 1 (Euler).
%e The table begins:
%e 3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, ...
%e 4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, ...
%e 5, 15, 215, 1285, 4435, 11310, 24490, 46610, 81005, 131560, 202610, 298690, ...
%e 6, 25, 397, 2461, 7873, 21271, 46183, 87475, 150445, 249985, 388885, 569839, ...
%e 7, 49, 1008, 5817, 19677, 49973, 106169, 200564, 346682, 560672, 861329, ...
%e 8, 65, 1601, 9225, 33201, 83361, 182705, 341433, 597169, 961761, 1490689, ...
%e 9, 144, 3069, 17316, 57555, 145062, 306684, 576783, 994230, 1605357, 2462112, ...
%e 10, 181, 4401, 25201, 87301, 218211, 469401, 877291, 1522231, 2452231, 3781541, ...
%e 11, 352, 7326, 40568, 133793, 335192, 706387, 1324851, 2279794, 3676431, ...
%e 12, 325, 7897, 53125, 182713, 456253, 990229, 1849549, 3207325, 5171497, ...
%e 13, 741, 14963, 81757, 267995, 668811, 1406366, 2632708, 4524910, ...
%e 14, 785, 19489, 107157, 360389, 893117, 1896665, 3536387, 6103889, ...
%e 15, 1395, 27420, 148335, 484005, 1204395, 2528445, 4726770, 8116650, ...
%e .
%e .
%e .
%Y Cf. A367303 (internal vertices), A367304 (regions), A367305 (edges), A366483 (first row), A367276 (second row).
%K nonn,tabl
%O 3,1
%A _Scott R. Shannon_, Nov 13 2023