A333283
Triangle read by rows: T(m,n) (m >= n >= 1) = number of edges formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.
Original entry on oeis.org
8, 28, 92, 80, 320, 1028, 178, 716, 2348, 5512, 372, 1604, 5332, 12676, 28552, 654, 2834, 9404, 22238, 49928, 87540, 1124, 5008, 16696, 39496, 88540, 156504, 279100, 1782, 7874, 26458, 62818, 141386, 251136, 447870
Offset: 1
Triangle begins:
8,
28, 92,
80, 320, 1028,
178, 716, 2348, 5512,
372, 1604, 5332, 12676, 28552,
654, 2834, 9404, 22238, 49928, 87540,
1124, 5008, 16696, 39496, 88540, 156504, 279100,
1782, 7874, 26458, 62818, 141386, 251136, 447870, ...
...
T(7,7) corrected Mar 19 2020
- Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
- Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
- Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
- Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
- Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
- Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
- N. J. A. Sloane, Illustration of T(3,2) = 320. [Black lines correspond to A331454(3,2), black + red lines correspond to A333278(3,2), and black + red + blue lines to T(3,2)]
- N. J. A. Sloane, Illustration of T(3,3) = 1028 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]
A333278
Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.
Original entry on oeis.org
8, 28, 92, 80, 296, 872, 178, 652, 1922, 4344, 372, 1408, 4256, 9738, 21284, 654, 2470, 7466, 16978, 36922, 64172, 1124, 4312, 13112, 29874, 64800, 113494, 200028, 1782, 6774, 20812, 47402, 103116, 181484, 319516, 509584, 2724, 10428, 31776, 72398, 158352, 279070, 490396, 782096, 1199428
Offset: 1
Triangle begins:
8,
28, 92,
80, 296, 872,
178, 652, 1922, 4344,
372, 1408, 4256, 9738, 21284,
654, 2470, 7466, 16978, 36922, 64172,
...
A347751
Number of finite edges in the graph formed when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.
Original entry on oeis.org
0, 8, 36, 124, 300, 664, 1200, 2108, 3388, 5232, 7568, 10852, 14892, 20288, 26704, 34540, 43812, 55400, 68584, 84684, 103004, 124216, 147888, 175820, 206788, 242424, 281560, 325708, 374148, 429416, 489000, 556412, 629804, 710536, 797280, 892564, 994588, 1107744, 1228432, 1359292, 1498788
Offset: 0
a(1) = 8 as connecting the four vertices of a single rectangle forms four new edges inside the rectangle, giving a total of 4 + 4 = 8 total edges.
a(2) = 36 as connecting the six vertices of two adjacent rectangles forms twenty-two edges inside the rectangles while also forming eight edges outside the rectangles. The total number of edges is then 6 + 22 + 8 = 36.
Cf.
A344993 (number of polygons),
A347750 (number of intersections),
A331757 (number of edges on or inside the rectangles).
A369177
Number of edges in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.
Original entry on oeis.org
6, 24, 66, 152, 318, 576, 998, 1608, 2474, 3600, 5162, 7100, 9670, 12772, 16546, 21036, 26622, 33024, 40814, 49716, 60006, 71560, 85158, 100264, 117626, 136780, 158358, 182080, 209106, 238312, 271314, 307304, 346866, 389488, 436286, 486444, 542026, 601436, 665814, 734504, 809882, 889544
Offset: 1
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).
Original entry on oeis.org
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
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.
Comments