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
KEYWORD
nonn
AUTHOR
Sascha Kurz, Jul 26 2004
STATUS
approved