A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

8, 28, 92, 80, 240, 360, 178, 508, 604, 860, 372, 944, 1040, 1320, 1792, 654, 1548, 1652, 1956, 2452, 3124, 1124, 2520, 2640, 2968, 3488, 4184, 5256, 1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188, 2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304, 3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460

Offset

1,1

Comments

See A331457 and A331776 for further illustrations.

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(3,3) = 360

Scott R. Shannon, Colored illustration for T(4,4) = 860

N. J. A. Sloane, Illustration for T(3,3) = 360.

Formula

Column 1 is A331757, for which there is an explicit formula.

Column 2 is A331765, for which no formula is known.

For m >= n >= 3, T(m,n) = (3*A332610(m,n)+4*A332611(m,n)+4*m+4*n-8)/2, and both A332610 and A332611 have explicit formulas.

Example

Triangle begins:
[8],
[28, 92],
[80, 240, 360],
[178, 508, 604, 860],
[372, 944, 1040, 1320, 1792],
[654, 1548, 1652, 1956, 2452, 3124],
[1124, 2520, 2640, 2968, 3488, 4184, 5256],
[1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188],
[2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304],
[3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460],
...

Crossrefs

Cf. A331457, A331757, A331765, A331776, A332599, A332610, A332611.

The main diagonal is A332597.

Sequence in context: A212516 A005343 A200941 * A331454 A333278 A333283

Adjacent sequences: A332597 A332598 A332599 * A332601 A332602 A332603

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Mar 03 2020

Extensions

More terms from N. J. A. Sloane, Mar 13 2020

Status

approved