A208454 -id:A208454 - cp's OEIS frontend

A097125 Number of noncongruent integer-sided tetrahedra with largest side n.

1, 4, 16, 45, 116, 254, 516, 956, 1669, 2760, 4379, 6676, 9888, 14219, 19956, 27421, 37062, 49143, 64272, 82888, 105629, 133132, 166090, 205223, 251624, 305861, 369247, 442695, 527417, 624483, 735777, 861885, 1005214, 1166797, 1348609

Offset: 1

Sascha Kurz, Jul 26 2004

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000 [Extracted from the Kurz link]
James East, Michael Hendriksen, and Laurence Park, On the enumeration of integer tetrahedra, arXiv:2112.00899 [math.CO], 2021.
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.
Sascha Kurz, Number of integral tetrahedra with given diameter, 2007.

Cf. A000065, A002620, A097126, A097127, A208454, A349295, A349296, A371344, A371345.

cmd3[d01_, d02_, d03_, d12_, d13_, d23s_] := Det[{{0, d01^2, d02^2, d03^2, 1}, {d01^2, 0, d12^2, d13^2, 1}, {d02^2, d12^2, 0, d23s, 1}, {d03^2, d13^2, d23s, 0, 1}, {1, 1, 1, 1, 0}}];
cmd30s = Sqrt /@ Solve[cmd3[d01, d02, d03, d12, d13, d23s] == 0, d23s][[;;,1,2]];
edgePermutations = PermutationList[#, 6] & /@ GroupElements@PermutationGroup[{Cycles[{{2, 4}, {3, 5}}], Cycles[{{1, 2}, {5, 6}}], Cycles[{{2, 3}, {4, 5}}]}];
canonical[dd_] := AllTrue[edgePermutations, OrderedQ[{dd[[#]], dd}] &];
a[d_] := Module[{s = 0, dd, uu}, Do[With[{roots = (cmd30s /. {d01 -> d})},
   dd = Min[Floor /@ roots + 1]; uu = Min[Max[Ceiling /@ roots - 1], d];
   Do[If[canonical[{d, d02, d03, d12, d13, d23}], s += 1], {d23, dd, uu}]],
  {d02, Quotient[d, 2] + 1, d}, {d12, d + 1 - d02, d02}, {d03, d + 1 - d02, d02}, {d13, d + 1 - d03, d02}]; s];
Array[a, 10] (* Andrey Zabolotskiy, Apr 04 2024, after Kurz's Algorithm 1 *)

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).

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.

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};

A371073 a(n)/144 is the maximum squared volume of a tetrahedron with the sum of its integer edge lengths equal to n.

Original entry on oeis.org

11, 14, 44, 128, 108, 188, 368, 448, 828, 1458, 1584, 2151, 3159, 3824, 5616, 8192, 9200, 11504, 15104, 17975, 23600, 31250, 35100, 41975, 51875, 60444, 74700, 93312, 104076, 120924, 143856, 164591, 195804, 235298, 260288, 296303, 343343, 387008, 448448, 524288

Offset: 9

Hugo Pfoertner, Mar 19 2024

nonn

			a(12) = 128 corresponds to the regular tetrahedron with all edges equal to 2. Its volume is V=sqrt(2)*2^3/12; V^2 = 2*2^6/12^2 = 128/144.

Cf. A097125, A208454, A371070, A371072.

Subset of A371071.

cp's OEIS Frontend

A097125 Number of noncongruent integer-sided tetrahedra with largest side 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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A371071 Squared volumes of tetrahedra with integer edge lengths, multiplied by 144.

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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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A371073 a(n)/144 is the maximum squared volume of a tetrahedron with the sum of its integer edge lengths equal to n.

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