A273464
The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.
Original entry on oeis.org
1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63
Offset: 1
Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
1;
1, 3;
1, 9, 24, 18;
1, 18, 126, 434, 762, 630, 187;
1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
...
- R. J. Mathar, Table of n, a(n) for n = 1..575
- J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.
- R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
- Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004.
- Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
- Wikipedia, Lozenge.
A326367
Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly two unit "lozenges" or "diamonds" (also of side length 1).
Original entry on oeis.org
0, 0, 24, 126, 387, 915, 1845, 3339, 5586, 8802, 13230, 19140, 26829, 36621, 48867, 63945, 82260, 104244, 130356, 161082, 196935, 238455, 286209, 340791, 402822, 472950, 551850, 640224, 738801, 848337, 969615, 1103445, 1250664, 1412136, 1588752, 1781430, 1991115
Offset: 1
We can represent a unit triangle this way:
o
/ \
o - o
and a unit "lozenge" or "diamond" has these three orientations:
o
/ \ o - o o - o
o o and / / and also \ \
\ / o - o o - o
o
and for n=3, here is one of the 24 different tiling of the triangle of side length 3 with exactly two lozenges:
o
/ \
o o
/ \ / \
o - o - o
/ / \ / \
o - o - o - o
- Colin Barker, Table of n, a(n) for n = 1..1000
- Richard J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
-
Rest@ CoefficientList[Series[3 x^3*(4 - x) (2 + x)/(1 - x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Jul 04 2019 *)
-
concat([0,0], Vec(3*x^3*(4 - x)*(2 + x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Jul 01 2019
A326368
Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly three unit "lozenges" or "diamonds" (also of side length 1).
Original entry on oeis.org
0, 0, 18, 434, 2814, 11127, 33365, 83568, 184254, 369254, 686952, 1203930, 2009018, 3217749, 4977219, 7471352, 10926570, 15617868, 21875294, 30090834, 40725702, 54318035, 71490993, 92961264, 119547974, 152182002, 191915700, 239933018, 297560034, 366275889
Offset: 1
We can represent a unit triangle this way:
o
/ \
o - o
and a unit "lozenge" or "diamond" has these three orientations:
o
/ \ o - o o - o
o o and / / and also \ \
\ / o - o o - o
o
and for n=3, here is one of the 18 different tiling of the triangle of side length 3 with exactly three lozenges:
o
/ \
o o
/ \ / \
o - o o
/ / \ / \
o - o - o - o
- Colin Barker, Table of n, a(n) for n = 1..1000
- Richard J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
-
Rest@ CoefficientList[Series[x^3*(18 + 308 x + 154 x^2 - 87 x^3 + 10 x^4 + 2 x^5)/(1 - x)^7, {x, 0, 30}], x] (* Michael De Vlieger, Jul 07 2019 *)
-
concat([0,0], Vec(x^3*(18 + 308*x + 154*x^2 - 87*x^3 + 10*x^4 + 2*x^5) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Jul 02 2019
A122722
Number of triangulations of Delta^2 x Delta^(k-1).
Original entry on oeis.org
1, 6, 108, 4488, 376200, 58652640, 16119956160, 7519632382080, 5788821019685760, 7197150396467808000, 14206044114169232371200, 43903287397136367836697600, 210012592354755890839147008000, 1540026232221309103088828327116800, 17170286302440610680613970557956096000, 289015112280462271460535463614055526400000
Offset: 1
a(1) = 1 * 1! = 1.
a(2) = 3 * 2! = 6.
a(3) = 18 * 3! = 108.
a(4) = "187 * 4! = 2244" [sic]; actually 187 * 4! = 4488.
a(5) = "3135 * 5! = 188100" [sic]; actually 3135 * 5! = 376200.
- J. A de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom., 15(3) (1996), 253-264. [It may be related to this sequence.]
- J. A. de Loera, J. Rambau, and Francisco Santos, MSRI Summer School on Triangulations of point sets, Applications, Structures and Algorithms.
- J. A. De Loera, J. Rambau, and Francisco Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), pp. 433-511.
- R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
- Francisco Santos, The Cayley trick, handwritten lecture notes; see table on p. 9.
- Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see Theorem 1 (p. 2).
- Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), pp. 151-177.
- Benjamin Frederik Schröter, Matroidal subdivisions, Dressians and tropical Grassmannians, Ph.D. Dissertation, Technische Universität Berlin, Berlin, 2018; see Appendix on p. 111.
A011555
Number of vertices of secondary polytope for triangle X n-simplex.
Original entry on oeis.org
6, 108, 4488, 376200
Offset: 1
- Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.
A011556
Number of regular triangulations of triangle X n-simplex.
Original entry on oeis.org
- Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.
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