A331457 - OEIS

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, 508, 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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..55.

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.

FORMULA