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Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of regions formed by straight line segments when connecting the k-1 points along the top side of a rectangle to each of the k-1 points along the bottom side that divide these sides into k equal parts, along with straight lines that directly connect the n-1 points along the left side to the diametrically opposite point on the right side that divide these sides into n equal parts.
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%I #32 Sep 08 2022 15:20:50

%S 1,2,2,6,4,3,18,10,6,4,48,24,16,8,5,106,56,34,20,10,6,216,116,70,44,

%T 26,12,7,382,228,134,84,58,30,14,8,650,396,250,152,112,60,36,16,9,

%U 1030,666,422,272,190,112,78,40,18,10,1564,1048,696,448,320,196,150,84,46,20,11

%N Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of regions formed by straight line segments when connecting the k-1 points along the top side of a rectangle to each of the k-1 points along the bottom side that divide these sides into k equal parts, along with straight lines that directly connect the n-1 points along the left side to the diametrically opposite point on the right side that divide these sides into n equal parts.

%H Scott R. Shannon, <a href="/A356790/a356790_1.txt">Table for n=1..40, k=1..40</a>.

%H Scott R. Shannon, <a href="/A356790/a356790.jpg">Image of T(3,4) = 34</a>.

%H Scott R. Shannon, <a href="/A356790/a356790_1.jpg">Image of T(5,7) = 320</a>.

%H Scott R. Shannon, <a href="/A356790/a356790_2.jpg">Image of T(8,6) = 256</a>.

%H Scott R. Shannon, <a href="/A356790/a356790_3.jpg">Image of T(11,14) = 5606</a>.

%H Scott R. Shannon, <a href="/A356790/a356790_4.jpg">Image of T(16,16) = 9964</a>.

%F T(1,k) = A306302(k-2) + 2, k >= 2.

%F T(2,k) = 2*A355902(k-2) + 4 = A306302(k-2) + 2*k, k >= 2.

%F T(n,1) = n.

%F T(n,2) = 2n.

%F T(n,3) = A146951(n).

%e The table begins:

%e 1, 2, 6, 18, 48, 106, 216, 382, 650, 1030, 1564, 2258, 3210, 4386, 5926, ...

%e 2, 4, 10, 24, 56, 116, 228, 396, 666, 1048, 1584, 2280, 3234, 4412, 5954, ...

%e 3, 6, 16, 34, 70, 134, 250, 422, 696, 1082, 1622, 2322, 3280, 4462, 6008, ...

%e 4, 8, 20, 44, 84, 152, 272, 448, 726, 1116, 1660, 2364, 3326, 4512, 6062, ...

%e 5, 10, 26, 58, 112, 190, 320, 506, 794, 1194, 1748, 2462, 3434, 4630, 6190, ...

%e 6, 12, 30, 60, 112, 196, 326, 512, 800, 1200, 1754, 2468, 3440, 4636, 6196, ...

%e 7, 14, 36, 78, 150, 258, 418, 626, 936, 1358, 1934, 2670, 3664, 4882, 6464, ...

%e 8, 16, 40, 84, 152, 256, 414, 632, 942, 1364, 1940, 2676, 3670, 4888, 6470, ...

%e 9, 18, 46, 94, 172, 290, 468, 710, 1050, 1490, 2084, 2838, 3850, 5086, 6686, ...

%e 10, 20, 50, 104, 188, 304, 480, 720, 1060, 1516, 2112, 2868, 3882, 5120, 6722, ...

%e 11, 22, 56, 118, 218, 366, 586, 878, 1280, 1794, 2454, 3258, 4320, 5606, 7256, ...

%e 12, 24, 60, 120, 208, 336, 518, 764, 1114, 1580, 2204, 2992, 4020, 5272, 6888, ...

%e .

%e .

%e See the attached table for further terms.

%Y Cf. A146951, A355798, A306302, A290131, A331452, A355902.

%K nonn,tabl

%O 1,2

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Sep 04 2022