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

4, 16, 56, 46, 142, 208, 104, 296, 348, 496, 214, 544, 592, 752, 1016, 380, 892, 948, 1120, 1396, 1784, 648, 1436, 1508, 1692, 1980, 2380, 2984, 1028, 2136, 2292, 2488, 2788, 3200, 3816, 4656, 1562, 3066, 3384, 3592, 3904, 4328, 4956, 5808, 6968, 2256, 4272, 4796, 5016, 5340, 5776, 6416, 7280, 8452, 9944

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Jan 29 2020

Keywords

• nonn
• tabl

Comments

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.

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.

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.

See A331776 for additional illustrations for the diagonal entries.

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.

Examples

			Triangle begins:
4,
16,56,
46,142,208,
104,296,348,496,
214,544,592,752,1016
380,892,948,1120,1396,1784
648,1436,1508,1692,1980,2380,2984
1028,2136,2292,2488,2788,3200,3816,4656
1562,3066,3384,3592,3904,4328,4956,5808,6968
2256,4272,4796,5016,5340,5776,6416,7280,8452,9944
		

Links

• Scott R. Shannon, Colored illustration for T(1,1) = 4.
• Scott R. Shannon, Colored illustration for T(2,2) = 56.
• Scott R. Shannon, Colored illustration for T(3,3) = 208.
• Scott R. Shannon, Colored illustration for T(4,4) = 496.
• Scott R. Shannon, Colored illustration for T(5,5) = 1016.
• Scott R. Shannon, Colored illustration for T(6,6) = 1784.
• Scott R. Shannon, Colored illustration for T(7,4) = 1692.
• Scott R. Shannon, Colored illustration for T(10,6) = 5776.
• N. J. A. Sloane, Illustration for T(3,3) = 208.

Crossrefs

Cf. A332599 (triangle giving numbers of vertices) and A332600 (edges).

Cf. also A331452.

The first column is A306302, the main diagonal is A331776.

Formula

Column 1 is A306302, for which there is an explicit formula.

Column 2 is A331766, for which no formula is known.

For n >= k >= 3, T(n,k) = A332610(n,k) + A332611(n,k), both of which have explicit formulas.

Extensions

More terms from Scott R. Shannon, Mar 05 2020

a(8) corrected by Giovanni Resta, May 22 2025