A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

5, 13, 37, 35, 99, 152, 75, 213, 256, 364, 159, 401, 448, 568, 776, 275, 657, 704, 836, 1056, 1340, 477, 1085, 1132, 1276, 1508, 1804, 2272, 755, 1619, 1712, 1868, 2112, 2420, 2900, 3532, 1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336, 1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516

Offset: 1

Author

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

Keywords

• nonn
• tabl

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.

Examples

			Triangle begins:
[5],
[13, 37],
[35, 99, 152],
[75, 213, 256, 364],
[159, 401, 448, 568, 776],
[275, 657, 704, 836, 1056, 1340],
[477, 1085, 1132, 1276, 1508, 1804, 2272],
[755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],
[1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],
[1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],
...
		

Links

• Scott R. Shannon, Colored illustration for T(3,3) = 152
• Scott R. Shannon, Colored illustration for T(4,4) = 364
• N. J. A. Sloane, Illustration for T(3,3) = 152.

Crossrefs

Cf. A331457, A331755, A331763, A331776, A332599, A332600.

The main diagonal is A332598.

Formula

Column 1 is A331755, for which there is an explicit formula.

Column 2 is A331763, for which no formula is known.

For m >= n >= 3, T(m,n) = A332600(m,n) - A331457(m,n) (Euler for genus 1 graph), and both A332600 and A331457 have explicit formulas.

Extensions

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