A019503
Simplexity of the n-cube: minimal cardinality of triangulation of n-cube using n-simplices whose vertices are vertices of the n-cube.
Original entry on oeis.org
1, 2, 5, 16, 67, 308, 1493
Offset: 1
- H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C9.
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.9, p. 512.
- Warren D. Smith, Lower bounds for triangulations of the N-cube, manuscript, 1994.
- Gunter M. Ziegler, Lectures on Polytopes, Revised First Edn., Graduate Texts in Mathematics, Springer, 1994, p. 147.
- A. Glazyrin, Lower bounds for the simplexity of the n-cube, Discrete Math. 312 (2012), no. 24, 3656--3662. MR2979495. - _N. J. A. Sloane_, Nov 07 2012
- R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discrete Mathematics, 158 (1996) 99-150, esp. p. 100.
- Mark Haiman, A simple and relatively efficient triangulation of the n-cube, Discrete Comput. Geometry 6 (1991), 287-289.
- D. Orden and F. Santos, Asymptotically efficient triangulations of the d-cube, Discr. Comput. Geom. 30 (2003) 509, Table 1.
- Warren D. Smith, A lower bound for the simplexity of the n-cube via hyperbolic volumes, Combinatorics of polytopes, European J. Combin. 21 (2000), no. 1, 131-137. MR1737333 (2001c:52004).
- Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.
A019504
Number of simplices in minimal corner-slicing triangulation of n-cube.
Original entry on oeis.org
1, 2, 5, 16, 67, 324, 1820
Offset: 1
- H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C9.
A275518
Number of simplices in corner-cut triangulation of the n-cube.
Original entry on oeis.org
1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1728604, 19011049, 228124384, 2965598547, 41518338684, 622774990133, 9964399645504, 169394793547567, 3049106282938684, 57933019373868897, 1158660387473183616, 24331868136927943019, 535301099012395872028
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Carl W. Lee, Triangulating the d-cube, Annals of the New York Academy of Sciences 440 (1985): 205-211.
- John F. Sallee, A note on minimal triangulations of an n-cube, Discrete Appl. Math. 4 (1982), no. 3, 211-215. MR0675850 (84g:52019)
- John F. Sallee, The middle-cut triangulations of the n-cube, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407-419. MR0752044 (86c:05054). See Table 2.
- John F. Sallee, A triangulation of the n-cube, Discrete Math. 40 (1982), no. 1, 81-86. MR0676714 (84d:05065b)
-
p := proc(d,x)
add( x^i/i!,i=0..d) ;
end proc:
A275518 := proc(d)
d!*(p(d,2)/2-p(d,1))+2^(d-1)-d!/2+1 ;
end proc:
seq(A275518(d),d=1..18) ;
-
p[d_, x_] := Sum[x^i/i!, {i, 0, d}];
A275518[d_] := d!*(p[d, 2]/2 - p[d, 1]) + 2^(d - 1) - d!/2 + 1;
Table[A275518[d], {d, 1, 18}] (* Jean-François Alcover, Sep 06 2023, after Maple program *)
-
a(n) = 1 + 2^(n-1) - n! + n!*sum(i=1, n, (2^(i-1)-1)/i!) \\ Andrew Howroyd, Sep 06 2023
A166932
Lower bounds for minimal number of simplices in a triangulation of the n-dimensional cube (A019503).
Original entry on oeis.org
5, 16, 67, 308, 1493, 5522
Offset: 3
- A. Bliss, F. E. Su, Lower bounds for simplicial covers and triangulations of cubes, arXiv:math/0310142 [math.CO], 2003 (see Table 1 page 4).
- A. Bliss, F. E. Su. Lower bounds for simplicial covers and triangulations of cubes, Discrete Comput. Geom. 33 (2005), 669-686.
- R. W. Cottle, Minimal triangulation of the 4-cube, Discrete Math., 40(1):25-29, 1982.
- Alexey Glazyrin, Lower bounds for the simplexity of the n-cube, arXiv:0910.4200 [math.MG], 2009-2012 (see Table 1 page 2).
- R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discrete Math., 158(1-3):99-150, 1996.
A239912
Number of simplices is middle-cut slicing of n-cube.
Original entry on oeis.org
1, 2, 5, 16, 67, 324, 1962, 13248, 106181, 931300
Offset: 1
Original entry on oeis.org
1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1720924
Offset: 1
- Sallee, John F. A note on minimal triangulations of an n-cube. Discrete Appl. Math. 4 (1982), no. 3, 211--215. MR0675850 (84g:52019)
- Sallee, John F. The middle-cut triangulations of the n-cube. SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407--419. MR0752044 (86c:05054). See Table 2.
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