A097125 -id:A097125 - cp's OEIS frontend

A208454 Number of nondegenerate noncongruent tetrahedra with integer edge lengths and total edge length n.

1, 0, 0, 1, 1, 1, 3, 2, 3, 6, 6, 7, 12, 11, 18, 21, 25, 31, 38, 46, 56, 66, 76, 90, 117, 123, 151, 175, 196, 234, 264, 297, 346, 391, 448, 492, 568, 630, 702, 797, 884, 977, 1089, 1217, 1338, 1469, 1624, 1771, 1970, 2146, 2343, 2579, 2782, 3042, 3322

Offset: 6

Philip Benjamin, Feb 26 2012

nonn

a(78) = 12345.

Selected values of a(n) up to n=20000 are contained in the East-Hendriksen-Park paper and supplementary website.

James East, Table of n, a(n) for n = 6..3000
James East, Michael Hendriksen, and Laurence Park, On the enumeration of integer tetrahedra, arXiv:2112.00899 [math.CO], 2021.
James East, Michael Hendriksen, and Laurence Park, On the enumeration of integer tetrahedra, supplementary material.

Cf. A097125.

A371070 a(n) is the number of distinct volumes > 0 of tetrahedra with the sum of their integer edge lengths equal to n.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 3, 6, 5, 7, 12, 10, 16, 19, 21, 26, 34, 37, 44, 56, 60, 67, 93, 92, 111, 137, 140, 166, 192, 211, 246, 279, 306, 333, 392, 428, 464, 538, 565, 627, 709, 768, 826, 939, 998, 1089, 1230, 1312, 1403, 1590, 1658, 1798, 1987, 2088, 2266, 2495

Offset: 6

Hugo Pfoertner, Mar 18 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 6..200
Hugo Pfoertner, Plot of ratio a(n)/A208454(n), using Plot 2. Is the asymptotic ratio for n->oo finite or 0?

Cf. A001402, A097125, A208454, A346575, A371071, A371345.

a371070(n) = {my (L=List()); forpart (w=n, forperm (w,v, if(v[4]+v[5]0, listput (L,CM))), [1,n], [6,6]); #Set(Vec(L))};

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
def A371070(n):
    CM = lambda x,y,z,t,u,v: (x*y*z<<2)+(a:=x+y-t)*(b:=x+z-u)*(c:=y+z-v)-x*c**2-y*b**2-z*a**2
    TR1 = lambda x,y,z: not(x+y0 and M not in d:
                    d.add(M)
                    c += 1
    return c # Chai Wah Wu, Mar 23 2024

a(n) <= A208454(n).

A371345 a(n) is the number of distinct volumes > 0 of tetrahedra with edges of integer length whose largest is n.

Original entry on oeis.org

1, 4, 16, 38, 96, 204, 424, 739, 1265, 2091, 3264, 4778, 7129, 10310, 14444, 19132, 26141, 34533, 44872, 57501, 73871, 93093, 114872, 139008, 175160, 211443, 255138, 306942, 364337, 431745, 506052, 586429, 696565, 803479, 948280, 1063150, 1226084, 1401161, 1606425, 1815322

Offset: 1

Hugo Pfoertner, Mar 19 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..112
Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.
Hugo Pfoertner, Plot of ratio a(n)/A097125(n) using Plot 2.
Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.

Cf. A097125, A349295, A371070, A371344.

\\ Cayley-Menger determinant
CM(v) = {matdet ([0,1,1,1,1; 1,0,v[1]^2,v[2]^2,v[3]^2; 1,v[1]^2,0,v[4]^2,v[5]^2; 1,v[2]^2,v[4]^2,0,v[6]^2; 1,v[3]^2,v[5]^2,v[6]^2,0])};
\\ First version using loops over 5 edges d_ij as described in Algorithm 1 (Sascha Kurz, 2008)
a371345(n) = {my (L=List(), v=vector(6)); v[1]=n; for (d02=floor((n+2)/2), n, v[2]=d02; for (d12=n+1-d02, d02, v[3]=d12; for (d03=n+1-d02, d02, v[4]=d03; for (d13=n+1-d03, d02, v[5]=d13; for (d23=1, n, v[6]=d23; forperm (v, w, my (c=CM(w)); if (c>0, listput(L, c)))))))); #Set(Vec(L))};
\\ Second version using simple minded loops and triangle inequalities. See Wirth
\\ and Dreiding (2009), p. 165, for justification to check only one triangle.
a371345(n) = {my (L=List(),w=vector(6)); w[1]=n; for(w2=1,n,w[2]=w2; for(w3=1,n,w[3]=w3; for(w4=1,n,w[4]=w4; for(w5=1,n,w[5]=w5; for(w6=1,n,w[6]=w6; forperm (w, v, if(v[4]+v[5]0, my(j=setsearch(L,c,1)); if (j>0, listinsert(~L,c,j))))))))); #Set(Vec(L))};

A371071 Squared volumes of tetrahedra with integer edge lengths, multiplied by 144.

Original entry on oeis.org

2, 11, 14, 26, 34, 44, 47, 54, 59, 62, 74, 98

Offset: 1

Hugo Pfoertner, Mar 18 2024

The larger terms depend on a lower bound for the minimum volume, which is not yet available. Therefore the data > 100 was removed. See A371072 for progress in determining this lower bound.

IBM Research, Tetrahedron Volumes, Ponder This Challenge November 2024.
Hugo Pfoertner, List of tetrahedra with small volumes, (2024); gives known further terms of the sequence.
Wikipedia, Cayley-Menger determinant
Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.

Cf. A097125, A208454, A371070, A371072, A371344.

A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.

Original entry on oeis.org

2, 11, 26, 47, 54, 107, 146, 191, 242, 299, 191, 134, 146, 146, 151, 767, 423, 151, 854, 558, 764, 491, 503, 464, 146, 146, 431, 944, 666, 146, 146, 350, 599, 311, 599, 511, 1719, 2286, 944, 1871, 1679, 990, 2714, 1907, 990, 551, 959, 1199, 1244, 990, 1206, 854, 764

Offset: 1

Hugo Pfoertner, Mar 19 2024

			a(1) = 2 corresponds to the regular tetrahedron with all edges equal to 1. Its volume is sqrt(2/144) = 0.11785113...

Hugo Pfoertner, Table of n, a(n) for n = 1..450
IBM Research, Tetrahedron Volumes, Ponder This Challenge November 2024.
Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.
Hugo Pfoertner, Plot of log_10(n) vs n, using Plot 2.

Cf. A097125, A371072, A371345.
Subset of A371071.
A001014(n)/72 are the corresponding maximum squared volumes.

\\ See A371345. Replace final #Set(Vec(L)) by vecmin(Vec(L))/2
\\ Second version using simple minded loops and triangle inequalities
\\ Not suitable for larger n
a371344(n) = {my (Vmin=oo,w=vector(6)); w[1]=n; for(w2=1,n,w[2]=w2; for(w3=1,n,w[3]=w3; for(w4=1,n,w[4]=w4; for(w5=1,n,w[5]=w5; for(w6=1,n,w[6]=w6;
forperm (w, v, if(v[4]+v[5]0, Vmin=min(Vmin,CM)))))))); Vmin/2}; \\ return value corrected by M. F. Hasler, Dec 02 2024

/* equivalent to the preceding, but simplified */
A371344(n) = {my (Vmin=oo,CM, n2=n^2); forvec(v=vector(5,k,[1,n]), v[4]+v[5]= Vmin || Vmin=CM); Vmin/2} \\ M. F. Hasler, Dec 02 2024

a(33), a(37), a(38), and a(43) corrected by Hugo Pfoertner, Dec 03 2024

A349295 a(n) is the number of ordered 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a tetrahedron ABCD with those edge-lengths, taken in a particular order (see comments).

Original entry on oeis.org

0, 1, 15, 124, 603, 2173, 6204, 15201, 33149, 66002, 122410, 214186, 357189, 572385, 886117, 1330930, 1947746, 2787431, 3907866, 5380602, 7288597, 9729060, 12815704, 16677303, 21461500, 27340308, 34501149, 43160975, 53560487, 65967718, 80677972, 98029728

Offset: 0

Giovanni Corbelli, Nov 13 2021

nonn

Edges with length a_1,a_2,a_3 form a face, a_1 is opposite to a_4, a_2 is opposite to a_5, a_3 is opposite to a_6. If the a_i's are all different, then there are 24 6-tuples corresponding to the same tetrahedron. The tetrahedron is possible iff triangular inequalities hold for every face and the Cayley-Menger determinant is positive. It has been proved that if triangular inequalities hold for at least one face and the Cayley-Menger determinant is positive, then the triangular inequalities for the other three faces hold, too (see article by Wirth, Dreiding in links, (5) at page 165).

Conjecture: The ratio a(n)/n^6 decreases with n and tends to a limit which is 0.10292439+-0,00000024 (1.96 sigmas, 95% confidence level) evaluated for n=2^32 on 6.4*10^12 random 6-tuples.

			For n=2 the 6-tuples are
(1,1,1,1,1,1),
(1,1,1,2,2,2), (1,2,2,2,1,1), (2,1,2,1,2,1), (2,2,1,1,1,2),
(2,2,1,2,2,1), (2,1,2,2,1,2), (1,2,2,1,2,2),
(1,2,2,2,2,2), (2,1,2,2,2,2), (2,2,1,2,2,2), (2,2,2,1,2,2), (2,2,2,2,1,2), (2,2,2,2,2,1),
(2,2,2,2,2,2)
corresponding to A097125(1) + A097125(2) = 5 different tetrahedra.

Giovanni Corbelli, Table of n, a(n) for n = 0..254
Giovanni Corbelli, FreeBasic program
Karl Wirth and André S. Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, Volume 64, Issue 4, 2009, pp. 160-170.

Cf. A097125, A349296, A346575.

A371072 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with the sum of its integer edge lengths equal to n.

Original entry on oeis.org

11, 14, 44, 26, 59, 34, 47, 126, 119, 62, 215, 54, 107, 98, 243, 146, 335, 142, 191, 614, 479, 194, 764, 423, 299, 254, 1004, 239, 851, 322, 304, 783, 887, 134, 479, 1719, 315, 234, 1196, 191, 896, 574, 767, 1127, 151, 674, 956, 956, 671, 146, 1391, 1082, 791, 898, 263, 1184, 151

Offset: 9

Hugo Pfoertner, Mar 19 2024

nonn

a(6) = 2, but since there are no tetrahedra with volume > 0 for n=7 and n=8, the offset 9 is chosen.

Hugo Pfoertner, Table of n, a(n) for n = 9..150

Cf. A097125, A208454, A371070, A371073, A371344.

Subset of A371071.

a371072(n) = {my (Vmin=oo); forpart (w=n, forperm (w, v, if(v[4]+v[5]0, Vmin=min(Vmin,CM))), [1, n], [6, 6]); Vmin/2};

A097126 Number of noncongruent integer-sided 4-dimensional simplices with largest side n.

Original entry on oeis.org

1, 6, 56, 336, 1840, 7925, 29183, 91621, 256546, 648697, 1508107, 3267671, 6679409, 12957976, 24015317, 42810244, 73793984, 123240964, 200260099, 317487746, 492199068, 747720800, 1115115145, 1634875673, 2360312092, 3358519981, 4716186332, 6541418450

Offset: 1

Sascha Kurz, Jul 26 2004

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..52 [Extracted from the Kurz link]
Sascha Kurz, Enumeration of integral tetrahedra, J. Integer Seqs., 10 (2007), # 07.9.3.
Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.

Cf. A002620, A097125, A097127.

A346575 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a tetrahedron ABCD with those edge-lengths.

Original entry on oeis.org

0, 1, 43, 327, 1792, 6139, 17607, 43291, 96142, 193149, 362383, 638533, 1075110, 1733023, 2700217, 4076133, 5994310, 8611819, 12119139, 16738861, 22746004, 30449013, 40212679, 52452031, 67651170, 86348035, 109166881, 136796079, 170024038, 209707144, 256814946, 312433795

Offset: 0

Giovanni Corbelli, Jul 24 2021

nonn

The existence of such a tetrahedron implies the following:
(1) there exists at least one permutation (a_i1,a_i2,a_i3,a_i4,a_i5,a_i6) such that triangular inequalities hold for (a_i1,a_i2,a_i3) (BCD), (a_i1,a_i4,a_i5) (ABC), (a_i2,a_i5,a_i6) (ACD) and (a_i3,a_i6,a_i4) (ABD), where we have a_i1=BC, a_i2=CD, a_i3=DB, a_i4=AB, a_i5=AC, a_i6=AD;
(2) a tetrahedron with such edge-lengths can be built.
Values were computed using a Visual Basic program with two different routines, manually checked for n = 2 and n = 3.
Conjecture 1: a(n)/n^6 tends to a limit which is 0.338170 +- 0.000017 (confidence level 95%). This number has been evaluated with a Monte-Carlo test on 3 billion sextuples with random values in (0,1) which simulate n -> oo.
Conjecture 2: there is no polynomial formula for a(n), as finite difference method fails.

			For a(2)=43 the solutions are (1,1,1,1,1,1), all 20 permutations of (1,1,1,2,2,2), all 15 permutations of (1,1,2,2,2,2), all 6 permutations of (1,2,2,2,2,2) and (2,2,2,2,2,2).

Chai Wah Wu, Table of n, a(n) for n = 0..52
Lucas A. Brown, Python program.
Giovanni Corbelli, Visual Basic routine generating number of tetrahedra.
Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.

Cf. A097125.

Equivalent sequence for triples with respect to triangles: A006003.

Python
```
# See LINKS.
```

Conjecture: Limit_{n->oo} a(n)/n^6 exists and is approximately 0.33817.

a(21)-a(31) from Lucas A. Brown, Mar 13 2024

A349296 First differences of A349295.

Original entry on oeis.org

1, 14, 109, 479, 1570, 4031, 8997, 17948, 32853, 56408, 91776, 143003, 215196, 313732, 444813, 616816, 839685, 1120435, 1472736, 1907995, 2440463, 3086644, 3861599, 4784197, 5878808, 7160841, 8659826, 10399512, 12407231, 14710254, 17351756

Offset: 1

Giovanni Corbelli, Nov 13 2021

nonn

a(n) is the number of ordered 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n}, with at least one element equal to n, such that there exists a tetrahedron ABCD with those edge-lengths, taken in a particular order (see comments in A349295).

Conjecture: for n tending to infinity the ratio a(n) / A097125(n) tends to 24 as the probability that all a_i's are different tends to 1 and there are 24 6-tuples corresponding to the same tetrahedron if all a_i's are different. For n=254 the ratio is 23.9936919.

Giovanni Corbelli, Table of n, a(n) for n = 1..254
Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.

Cf. A097125, A349295.

cp's OEIS Frontend

A208454 Number of nondegenerate noncongruent tetrahedra with integer edge lengths and total edge length n.

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A371070 a(n) is the number of distinct volumes > 0 of tetrahedra with the sum of their integer edge lengths equal to n.

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A371345 a(n) is the number of distinct volumes > 0 of tetrahedra with edges of integer length whose largest is n.

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A371071 Squared volumes of tetrahedra with integer edge lengths, multiplied by 144.

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A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.

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A349295 a(n) is the number of ordered 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a tetrahedron ABCD with those edge-lengths, taken in a particular order (see comments).

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A371072 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with the sum of its integer edge lengths equal to n.

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PARI

A097126 Number of noncongruent integer-sided 4-dimensional simplices with largest side n.

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A346575 a(n) is the number of 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a tetrahedron ABCD with those edge-lengths.

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A349296 First differences of A349295.