Search: a349295 -id:a349295
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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
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1,2
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COMMENTS
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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.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A097125
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Number of noncongruent integer-sided tetrahedra with largest side n.
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+10
13
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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
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1,2
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A371345
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a(n) is the number of distinct volumes > 0 of tetrahedra with edges of integer length whose largest is n.
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+10
6
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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
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OFFSET
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1,2
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LINKS
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PROG
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(PARI) \\ 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]<v[6], next); if(v[4]+v[6]<v[5], next); if(v[5]+v[6]<v[4], next); my(c=CM(v)); if (c>0, my(j=setsearch(L, c, 1)); if (j>0, listinsert(~L, c, j))))))))); #Set(Vec(L))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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