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

%I A097125 #42 Apr 05 2024 06:29:00
%S A097125 1,4,16,45,116,254,516,956,1669,2760,4379,6676,9888,14219,19956,27421,
%T A097125 37062,49143,64272,82888,105629,133132,166090,205223,251624,305861,
%U A097125 369247,442695,527417,624483,735777,861885,1005214,1166797,1348609
%N A097125 Number of noncongruent integer-sided tetrahedra with largest side n.
%H A097125 Andrey Zabolotskiy, <a href="/A097125/b097125.txt">Table of n, a(n) for n = 1..1000</a> [Extracted from the Kurz link]
%H A097125 James East, Michael Hendriksen, and Laurence Park, <a href="https://arxiv.org/abs/2112.00899">On the enumeration of integer tetrahedra</a>, arXiv:2112.00899 [math.CO], 2021.
%H A097125 Sascha Kurz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kurz/kurz3.html">Enumeration of integral tetrahedra</a>, J. Integer Seqs., 10 (2007), # 07.9.3.
%H A097125 Sascha Kurz, <a href="http://arXiv.org/abs/0804.1310">Enumeration of integral tetrahedra</a>, arXiv:0804.1310 [math.CO], 2008.
%H A097125 Sascha Kurz, <a href="https://web.archive.org/web/20070808064907/https://www.wm.uni-bayreuth.de/index.php?id=252">Number of integral tetrahedra with given diameter</a>, 2007.
%t A097125 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}}];
%t A097125 cmd30s = Sqrt /@ Solve[cmd3[d01, d02, d03, d12, d13, d23s] == 0, d23s][[;;,1,2]];
%t A097125 edgePermutations = PermutationList[#, 6] & /@ GroupElements@PermutationGroup[{Cycles[{{2, 4}, {3, 5}}], Cycles[{{1, 2}, {5, 6}}], Cycles[{{2, 3}, {4, 5}}]}];
%t A097125 canonical[dd_] := AllTrue[edgePermutations, OrderedQ[{dd[[#]], dd}] &];
%t A097125 a[d_] := Module[{s = 0, dd, uu}, Do[With[{roots = (cmd30s /. {d01 -> d})},
%t A097125    dd = Min[Floor /@ roots + 1]; uu = Min[Max[Ceiling /@ roots - 1], d];
%t A097125    Do[If[canonical[{d, d02, d03, d12, d13, d23}], s += 1], {d23, dd, uu}]],
%t A097125   {d02, Quotient[d, 2] + 1, d}, {d12, d + 1 - d02, d02}, {d03, d + 1 - d02, d02}, {d13, d + 1 - d03, d02}]; s];
%t A097125 Array[a, 10] (* _Andrey Zabolotskiy_, Apr 04 2024, after Kurz's Algorithm 1 *)
%Y A097125 Cf. A000065, A002620, A097126, A097127, A208454, A349295, A349296, A371344, A371345.
%K A097125 nonn
%O A097125 1,2
%A A097125 _Sascha Kurz_, Jul 26 2004