A371070 -id:A371070 - cp's OEIS frontend

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

Hugo Pfoertner, Mar 19 2024

nonn

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.

Cf. A097125, A349295, A371070, A371344.

\\ 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))};

A371071 Squared volumes of tetrahedra with integer edge lengths, multiplied by 144.

Original entry on oeis.org

2, 11, 14, 26, 34, 44, 47, 54, 59, 62, 74, 98

Offset: 1

Hugo Pfoertner, Mar 18 2024

The larger terms depend on a lower bound for the minimum volume, which is not yet available. Therefore the data > 100 was removed. See A371072 for progress in determining this lower bound.

IBM Research, Tetrahedron Volumes, Ponder This Challenge November 2024.
Hugo Pfoertner, List of tetrahedra with small volumes, (2024); gives known further terms of the sequence.
Wikipedia, Cayley-Menger determinant
Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.

Cf. A097125, A208454, A371070, A371072, A371344.

A371072 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with the sum of its integer edge lengths equal to n.

Original entry on oeis.org

11, 14, 44, 26, 59, 34, 47, 126, 119, 62, 215, 54, 107, 98, 243, 146, 335, 142, 191, 614, 479, 194, 764, 423, 299, 254, 1004, 239, 851, 322, 304, 783, 887, 134, 479, 1719, 315, 234, 1196, 191, 896, 574, 767, 1127, 151, 674, 956, 956, 671, 146, 1391, 1082, 791, 898, 263, 1184, 151

Offset: 9

Hugo Pfoertner, Mar 19 2024

nonn

a(6) = 2, but since there are no tetrahedra with volume > 0 for n=7 and n=8, the offset 9 is chosen.

Hugo Pfoertner, Table of n, a(n) for n = 9..150

Cf. A097125, A208454, A371070, A371073, A371344.

Subset of A371071.

a371072(n) = {my (Vmin=oo); forpart (w=n, forperm (w, v, if(v[4]+v[5]0, Vmin=min(Vmin,CM))), [1, n], [6, 6]); Vmin/2};

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Hugo Pfoertner, Apr 16 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};

def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

b-file corrected by David Radcliffe, Aug 01 2025

A371073 a(n)/144 is the maximum squared volume of a tetrahedron with the sum of its integer edge lengths equal to n.

Original entry on oeis.org

11, 14, 44, 128, 108, 188, 368, 448, 828, 1458, 1584, 2151, 3159, 3824, 5616, 8192, 9200, 11504, 15104, 17975, 23600, 31250, 35100, 41975, 51875, 60444, 74700, 93312, 104076, 120924, 143856, 164591, 195804, 235298, 260288, 296303, 343343, 387008, 448448, 524288

Offset: 9

Hugo Pfoertner, Mar 19 2024

nonn

			a(12) = 128 corresponds to the regular tetrahedron with all edges equal to 2. Its volume is V=sqrt(2)*2^3/12; V^2 = 2*2^6/12^2 = 128/144.

Cf. A097125, A208454, A371070, A371072.

Subset of A371071.

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

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A371071 Squared volumes of tetrahedra with integer edge lengths, multiplied by 144.

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A371072 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with the sum of its integer edge lengths equal to n.

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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

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A371073 a(n)/144 is the maximum squared volume of a tetrahedron with the sum of its integer edge lengths equal to n.

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