A019503 -id:A019503 - cp's OEIS frontend

A166932 Lower bounds for minimal number of simplices in a triangulation of the n-dimensional cube (A019503).

5, 16, 67, 308, 1493, 5522

Offset: 3

Jonathan Vos Post, Oct 24 2009

The terms are given in Table 1 on page 2 of the Glazyrin reference.

There are many lists of bounds in different papers which differ by range, values, and methods used to obtain them. - Andrey Zabolotskiy, Nov 17 2017

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.

Cf. A019502, A019503, A019504.

A019502 Number of simplices in minimal decomposition of an n-cube.

Original entry on oeis.org

1, 2, 5, 16, 67

Offset: 1

N. J. A. Sloane

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, Section C9.
R. K. Guy, What is the simplexity of the d-cube?, Amer. Math. Monthly, 91:10 (1984), 628-629.

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. B. Hughes and M. R. Anderson, Simplexity of the cube, Discrete Math., 158(1-3):99-150, 1996.
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).

Other sequences dealing with different ways to attack this problem. They give further references: A019503, A019504, A166932, A166932, A239912, A275518.

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

N. J. A. Sloane

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C9.

R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discr. Math., 158 (1996), 99-150.

Cf. A019502, A019503.

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

R. J. Mathar, Jul 31 2016

This corrects the value of a(10) in A239911 published by Sallee in Discr. Math. 40. The correct value is for example given by Lee.

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)

Cf. A239911, A019502, A019503, A019504, A166932.

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

a(n) = 1 + 2^(n-1) - n! + n!*Sum_{i=1..n} (2^(i-1)-1)/i!. - Andrew Howroyd, Sep 06 2023, after Maple program

Terms a(19) and beyond from Andrew Howroyd, Sep 06 2023

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

N. J. A. Sloane, Apr 09 2014

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).

Other sequences dealing with different ways to attack this problem. They give further references: A019502, A019503, A019504, A166932, A166932, A275518.

A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

Original entry on oeis.org

1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742

Offset: 0

Jonathan Vos Post, Jul 15 2008

Andrew Weimholt suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.

			a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.

Georg Fischer, Table of n, a(n) for n = 0..1000
Gil Kalai, Five Open Problems Regarding Convex Polytopes.
Eric W. Weisstein, Polytope
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000943, A000944, A019503, A053016, A060296, A063924, A063925, A063926, A063927, A065984, A086653, A093478, A093479, A105230, A105231.

LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)

For n > 4, a(n) = A086653(n) + 1 = 2^n + 3*n + 1.

G.f.: -(1488*x^7 - 3656*x^6 + 2794*x^5 - 569*x^4 - 58*x^3 + 3*x^2 + 2*x - 1)/((1-x)^2*(1-2*x)). [Colin Barker, Sep 05 2012]

a(14)-a(15) corrected by Georg Fischer, May 02 2019

A239911 Erroneous version of A275518.

Original entry on oeis.org

1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1720924

Offset: 1

N. J. A. Sloane, Apr 09 2014

dead

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.

Carl W. Lee, Triangulating the d-cube, Annals of the New York Academy of Sciences 440 (1985): 205-211.
Sallee, John F. A triangulation of the n-cube, Discrete Math. 40 (1982), no. 1, 81--86. MR0676714 (84d:05065b)

Cf. A019502, A019503, A019504, A166932.

A243311 Decimal expansion of the volume of a regular ideal hyperbolic 4-simplex.

Original entry on oeis.org

2, 6, 8, 8, 9, 5, 6, 6, 0, 1, 6, 9, 3, 1, 1, 2, 2, 5, 4, 6, 9, 2, 9, 4, 8, 6, 7, 2, 2, 7, 2, 4, 5, 6, 6, 4, 5, 2, 4, 9, 9, 4, 4, 3, 7, 1, 9, 3, 0, 3, 1, 3, 7, 2, 9, 2, 1, 0, 7, 6, 4, 8, 0, 2, 5, 7, 6, 3, 9, 2, 6, 0, 9, 0, 0, 9, 9, 2, 6, 4, 7, 2, 9, 4, 4, 3, 7, 4, 8, 1, 3, 4, 8, 4, 3, 1, 8, 1, 9, 7, 5, 3

Offset: 0

Jean-François Alcover, Jun 03 2014

			0.268895660169311225469294867227245664524994437...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.9 Hyperbolic volume constants, p. 512.

Cf. A019503, A143298 (Gieseking's constant is also the volume of a regular ideal hyperbolic 3-simplex).

RealDigits[10*Pi/3*ArcSin[1/3] - Pi^2/3, 10, 102] // First

10*Pi/3*asin(1/3) - Pi^2/3 \\ Stefano Spezia, Dec 24 2024

Equals 10*Pi/3*arcsin(1/3) - Pi^2/3.

cp's OEIS Frontend

A166932 Lower bounds for minimal number of simplices in a triangulation of the n-dimensional cube (A019503).

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A019502 Number of simplices in minimal decomposition of an n-cube.

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A019504 Number of simplices in minimal corner-slicing triangulation of n-cube.

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A275518 Number of simplices in corner-cut triangulation of the n-cube.

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A239912 Number of simplices is middle-cut slicing of n-cube.

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A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

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Programs

Mathematica

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A239911 Erroneous version of A275518.

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Links

Crossrefs

A243311 Decimal expansion of the volume of a regular ideal hyperbolic 4-simplex.

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Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula