A335701 Irregular triangle read by rows: consider the structure formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of an (n+1) X 3 rectangular grid of points (or equally, an n X 2 grid of squares); row n gives number of cells with k sides, for k >= 3.

14, 2, 48, 8, 102, 36, 4, 192, 92, 12, 326, 194, 24, 524, 336, 28, 4, 802, 554, 80, 1192, 812, 128, 4, 1634, 1314, 112, 0, 4, 2, 2296, 1756, 200, 20, 3074, 2508, 236, 22, 4052, 3252, 356, 28, 5246, 4348, 472, 28, 6740, 5464, 652, 28, 8398, 7054, 656, 74, 10440, 8760, 940, 52, 12770, 11050, 1040, 58, 15512, 13324, 1300, 60, 4, 18782, 16162, 1600, 70, 22384, 19256, 1948, 104

Offset

1,1

Comments

More than the usual number of terms are given, in order to include the first 20 rows and emphasize the fact that so far k is never more than 8.

These are the structures discussed in column 2 of the table in A331452. It is known that the structures discussed in column 1 of that table have cells with at most 4 sides, so an upper limit of 8 sides for the present sequence is certainly possible.

The maximum number of sides for n=19..106 is 6. - Lars Blomberg, Aug 27 2020

Links

Lars Blomberg, Table of n, a(n) for n = 1..419 (the first 106 rows)

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

Scott R. Shannon, Colored illustration for n=1

Scott R. Shannon, Colored illustration for n=2

Scott R. Shannon, Colored illustration for n=3

Scott R. Shannon, Colored illustration for n=4

Scott R. Shannon, Colored illustration for n=5

Scott R. Shannon, Colored illustration for n=6

Scott R. Shannon, Colored illustration for n=7

Scott R. Shannon, Colored illustration for n=8

Scott R. Shannon, Colored illustration for n=9

Scott R. Shannon, Colored illustration for n=10

Scott R. Shannon, Colored illustration for n=11

Scott R. Shannon, Colored illustration for n=12

Scott R. Shannon, Colored illustration for n=13

Scott R. Shannon, Colored illustration for n=14

Scott R. Shannon, Colored illustration for n=15

Scott R. Shannon, Colored illustration for n=16

Scott R. Shannon, Colored illustration for n=2. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=3. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=4. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=5. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=6. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=7. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=8. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=9. Random distance-based coloring.

Scott R. Shannon, Colored illustration for n=10. Random distance-based coloring.

Example

Triangle begins:
14, 2,
48, 8,
102, 36, 4,
192, 92, 12
326, 194, 24
524, 336, 28, 4
802, 554, 80,
1192, 812, 128, 4
1634, 1314, 112, 0, 4, 2
2296, 1756, 200, 20
3074, 2508, 236, 22
4052, 3252, 356, 28
5246, 4348, 472, 28
6740, 5464, 652, 28
8398, 7054, 656, 74
10440, 8760, 940, 52
12770, 11050, 1040, 58
15512, 13324, 1300, 60, 4
18782, 16162, 1600, 70
22384, 19256, 1948, 104
...
The 1X2 structure (or 2X1 structure, as in the illustration) contains 14 triangles and 2 quadrilaterals, so row 1 is 14, 2.
The 3X2 structure contains 102 triangles, 36 quadrilaterals, and 4 pentagons, so row 3 is 102, 36, 4. The sum is 142 = A331766(3).

Crossrefs

Cf. A331452, A331766 (row sums), A331763, A331765.

Sequence in context: A067740 A037923 A040195 * A303380 A242061 A040189

Adjacent sequences: A335698 A335699 A335700 * A335702 A335703 A335704

Keyword

nonn,tabf

Author

Scott R. Shannon and N. J. A. Sloane, Aug 09 2020

Status

approved