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

1,2

Links

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.

Mathematica

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 *)

Crossrefs

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

Sequence in context: A359073 A134139 A213501 * A213480 A306302 A159940

Adjacent sequences: A097122 A097123 A097124 * A097126 A097127 A097128

Keyword

nonn

Author

Sascha Kurz, Jul 26 2004

Status

approved