A336731 Three-column table read by rows: row n gives [number of triangle-triangle, triangle-quadrilateral, quadrilateral-quadrilateral] contacts for a row of n adjacent congruent rectangles divided by drawing diagonals of all possible rectangles (cf. A331452).

4, 0, 0, 14, 8, 0, 20, 48, 4, 60, 80, 28, 68, 224, 68, 148, 368, 124, 224, 616, 268, 336, 1008, 420, 384, 1672, 648, 712, 2208, 972, 972, 3120, 1464, 1300, 4304, 1996, 1496, 6040, 2788, 2044, 7936, 3580, 2612, 10224, 4672, 3540, 12656, 5980, 4224, 16104, 7676, 5484, 19648, 9500

Offset: 1

Author

Scott R. Shannon, Aug 02 2020

Keywords

• nonn
• tabf

Comments

For a row of n adjacent rectangles the only polygons formed when dividing all possible rectangles along their diagonals are 3-gons (triangles) and 4-gons (quadrilaterals). Hence the only possible edge-sharing contacts are 3-gons with 3-gons, 3-gons with 4-gons, and 4-gons with 4-gons. This sequence lists the number of these three possible combinations for a row of n adjacent rectangles. Note that the edges along the outside of the n adjacent rectangles are not counted as they are only in one n-gon.

These are graphs T(1,n) described in A331452. - N. J. A. Sloane, Aug 03 2020

Examples

			a(1) = 4, a(2) = 0, a(3) = 0. A single rectangle divided along its diagonals consists of four 3-gons, four edges, and no 4-gons. Therefore there are only four 3-gon-to-3-gon contacts. See the link image for n = 1.
a(4) = 14, a(5) = 8, a(6) = 0. Two adjacent rectangles divided along all diagonals consists of fourteen 3-gons and two 4-gons. The two 4-gons are separated and thus share all their edges, eight in total, with 3-gons. There are fourteen pairs of 3-gon-to-3-gon contacts. See the link image for n = 2.
a(7) = 20, a(8) = 48, a(9) = 4. Three adjacent rectangles divided along all diagonals consists of thirty-two 3-gons and fourteen 4-gons. There are two groups of three adjacent 4-gons, so there are four 4-gons-to-4-gon contacts. These, along with the other 4-gons, share 48 edges with 3-gons. There are also twenty 3-gon-to-3-gon contacts. See the link image for n = 3.
.
The table begins:
4,0,0;
14,8,0;
20,48,4;
60,80,28;
68,224,68;
148,368,124;
224,616,268;
336,1008,420;
384,1672,648;
712,2208,972;
972,3120,1464;
1300,4304,1996;
1496,6040,2788;
2044,7936,3580;
2612,10224,4672;
3540,12656,5980;
4224,16104,7676;
5484,19648,9500;
6568,24216,11936;
7836,29616,14468;
See A306302 for a count of the regions and images for other values of n.
		

Links

• Scott R. Shannon, Image of the rectangles for n = 1.
• Scott R. Shannon, Image of the rectangles for n = 2.
• Scott R. Shannon, Image of the rectangles for n = 3.
• Scott R. Shannon, Image of the rectangles for n = 4.

Crossrefs

Cf. A306302, A331452, A331755, A331757, A333288.

Formula

Sum of row t = A331757(t) - 2(t + 1).