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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).
2
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
1,1
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
FORMULA
Sum of row t = A331757(t) - 2(t + 1).
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Scott R. Shannon, Aug 02 2020
STATUS
approved