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A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

%I A331457 #60 May 22 2025 03:31:15
%S A331457 4,16,56,46,142,208,104,296,348,496,214,544,592,752,1016,380,892,948,
%T A331457 1120,1396,1784,648,1436,1508,1692,1980,2380,2984,1028,2136,2292,2488,
%U A331457 2788,3200,3816,4656,1562,3066,3384,3592,3904,4328,4956,5808,6968,2256,4272,4796,5016,5340,5776,6416,7280,8452,9944
%N A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).
%C A331457 A "frame" of size n X k is formed from a grid of (n+1) X (k+1) points with the central grid of (n-3) X (k-3) points removed. If n or k is less than 3 then no points are removed, and T(n,k) = A331452(n,k). From now on we assume both n and k are >= 3.
%C A331457 The resulting array has an outer perimeter with 2*(n+k) points and an inner perimeter with 2*(n+k)-8 points, for a total of 4*(n+k)-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.
%C A331457 Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
%C A331457 See A331776 for additional illustrations for the diagonal entries.
%C A331457 There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.
%H A331457 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for T(1,1) = 4.</a>
%H A331457 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for T(2,2) = 56.</a>
%H A331457 Scott R. Shannon, <a href="/A331776/a331776.png">Colored illustration for T(3,3) = 208.</a>
%H A331457 Scott R. Shannon, <a href="/A331776/a331776_1.png">Colored illustration for T(4,4) = 496.</a>
%H A331457 Scott R. Shannon, <a href="/A331776/a331776_2.png">Colored illustration for T(5,5) = 1016.</a>
%H A331457 Scott R. Shannon, <a href="/A331776/a331776_3.png">Colored illustration for T(6,6) = 1784.</a>
%H A331457 Scott R. Shannon, <a href="/A331457/a331457.png">Colored illustration for T(7,4) = 1692.</a>
%H A331457 Scott R. Shannon, <a href="/A331457/a331457_1.png">Colored illustration for T(10,6) = 5776.</a>
%H A331457 N. J. A. Sloane, <a href="/A331457/a331457.pdf">Illustration for T(3,3) = 208.</a>
%F A331457 Column 1 is A306302, for which there is an explicit formula.
%F A331457 Column 2 is A331766, for which no formula is known.
%F A331457 For n >= k >= 3, T(n,k) = A332610(n,k) + A332611(n,k), both of which have explicit formulas.
%e A331457 Triangle begins:
%e A331457 4,
%e A331457 16,56,
%e A331457 46,142,208,
%e A331457 104,296,348,496,
%e A331457 214,544,592,752,1016
%e A331457 380,892,948,1120,1396,1784
%e A331457 648,1436,1508,1692,1980,2380,2984
%e A331457 1028,2136,2292,2488,2788,3200,3816,4656
%e A331457 1562,3066,3384,3592,3904,4328,4956,5808,6968
%e A331457 2256,4272,4796,5016,5340,5776,6416,7280,8452,9944
%Y A331457 Cf. A332599 (triangle giving numbers of vertices) and A332600 (edges).
%Y A331457 Cf. also A331452.
%Y A331457 The first column is A306302, the main diagonal is A331776.
%K A331457 nonn,tabl
%O A331457 1,1
%A A331457 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 29 2020
%E A331457 More terms from _Scott R. Shannon_, Mar 05 2020
%E A331457 a(8) corrected by _Giovanni Resta_, May 22 2025