A371345 a(n) is the number of distinct volumes > 0 of tetrahedra with edges of integer length whose largest is n.

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

Offset: 1

Author

Hugo Pfoertner, Mar 19 2024

Keywords

• nonn

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..112
• Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.
• Hugo Pfoertner, Plot of ratio a(n)/A097125(n) using Plot 2.
• Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.

Crossrefs

Cf. A097125, A349295, A371070, A371344.

Programs

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]0, my(j=setsearch(L,c,1)); if (j>0, listinsert(~L,c,j))))))))); #Set(Vec(L))};