cp's OEIS Frontend

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).

%I A336731 #30 Aug 03 2020 10:05:11
%S A336731 4,0,0,14,8,0,20,48,4,60,80,28,68,224,68,148,368,124,224,616,268,336,
%T A336731 1008,420,384,1672,648,712,2208,972,972,3120,1464,1300,4304,1996,1496,
%U A336731 6040,2788,2044,7936,3580,2612,10224,4672,3540,12656,5980,4224,16104,7676,5484,19648,9500
%N 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).
%C A336731 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.
%C A336731 These are graphs T(1,n) described in A331452. - _N. J. A. Sloane_, Aug 03 2020
%H A336731 Scott R. Shannon, <a href="/A336731/a336731_3.png">Image of the rectangles for n = 1</a>.
%H A336731 Scott R. Shannon, <a href="/A336731/a336731.png">Image of the rectangles for n = 2</a>.
%H A336731 Scott R. Shannon, <a href="/A336731/a336731_1.png">Image of the rectangles for n = 3</a>.
%H A336731 Scott R. Shannon, <a href="/A336731/a336731_2.png">Image of the rectangles for n = 4</a>.
%F A336731 Sum of row t = A331757(t) - 2(t + 1).
%e A336731 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.
%e A336731 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.
%e A336731 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.
%e A336731 .
%e A336731 The table begins:
%e A336731 4,0,0;
%e A336731 14,8,0;
%e A336731 20,48,4;
%e A336731 60,80,28;
%e A336731 68,224,68;
%e A336731 148,368,124;
%e A336731 224,616,268;
%e A336731 336,1008,420;
%e A336731 384,1672,648;
%e A336731 712,2208,972;
%e A336731 972,3120,1464;
%e A336731 1300,4304,1996;
%e A336731 1496,6040,2788;
%e A336731 2044,7936,3580;
%e A336731 2612,10224,4672;
%e A336731 3540,12656,5980;
%e A336731 4224,16104,7676;
%e A336731 5484,19648,9500;
%e A336731 6568,24216,11936;
%e A336731 7836,29616,14468;
%e A336731 See A306302 for a count of the regions and images for other values of n.
%Y A336731 Cf. A306302, A331452, A331755, A331757, A333288.
%K A336731 nonn,tabf
%O A336731 1,1
%A A336731 _Scott R. Shannon_, Aug 02 2020