A371345 -id:A371345 - cp's OEIS frontend

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

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

Offset: 1

Sascha Kurz, Jul 26 2004

nonn

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.

Cf. A000065, A002620, A097126, A097127, A208454, A349295, A349296, A371344, A371345.

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 *)

A371070 a(n) is the number of distinct volumes > 0 of tetrahedra with the sum of their integer edge lengths equal to n.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 3, 6, 5, 7, 12, 10, 16, 19, 21, 26, 34, 37, 44, 56, 60, 67, 93, 92, 111, 137, 140, 166, 192, 211, 246, 279, 306, 333, 392, 428, 464, 538, 565, 627, 709, 768, 826, 939, 998, 1089, 1230, 1312, 1403, 1590, 1658, 1798, 1987, 2088, 2266, 2495

Offset: 6

Hugo Pfoertner, Mar 18 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 6..200
Hugo Pfoertner, Plot of ratio a(n)/A208454(n), using Plot 2. Is the asymptotic ratio for n->oo finite or 0?

Cf. A001402, A097125, A208454, A346575, A371071, A371345.

a371070(n) = {my (L=List()); forpart (w=n, forperm (w,v, if(v[4]+v[5]0, listput (L,CM))), [1,n], [6,6]); #Set(Vec(L))};

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_permutations
def A371070(n):
    CM = lambda x,y,z,t,u,v: (x*y*z<<2)+(a:=x+y-t)*(b:=x+z-u)*(c:=y+z-v)-x*c**2-y*b**2-z*a**2
    TR1 = lambda x,y,z: not(x+y0 and M not in d:
                    d.add(M)
                    c += 1
    return c # Chai Wah Wu, Mar 23 2024

a(n) <= A208454(n).

A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.

Original entry on oeis.org

2, 11, 26, 47, 54, 107, 146, 191, 242, 299, 191, 134, 146, 146, 151, 767, 423, 151, 854, 558, 764, 491, 503, 464, 146, 146, 431, 944, 666, 146, 146, 350, 599, 311, 599, 511, 1719, 2286, 944, 1871, 1679, 990, 2714, 1907, 990, 551, 959, 1199, 1244, 990, 1206, 854, 764

Offset: 1

Hugo Pfoertner, Mar 19 2024

			a(1) = 2 corresponds to the regular tetrahedron with all edges equal to 1. Its volume is sqrt(2/144) = 0.11785113...

Hugo Pfoertner, Table of n, a(n) for n = 1..450
IBM Research, Tetrahedron Volumes, Ponder This Challenge November 2024.
Sascha Kurz, Enumeration of integral tetrahedra, arXiv:0804.1310 [math.CO], 2008.
Hugo Pfoertner, Plot of log_10(n) vs n, using Plot 2.

Cf. A097125, A371072, A371345.
Subset of A371071.
A001014(n)/72 are the corresponding maximum squared volumes.

\\ See A371345. Replace final #Set(Vec(L)) by vecmin(Vec(L))/2
\\ Second version using simple minded loops and triangle inequalities
\\ Not suitable for larger n
a371344(n) = {my (Vmin=oo,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, Vmin=min(Vmin,CM)))))))); Vmin/2}; \\ return value corrected by M. F. Hasler, Dec 02 2024

/* equivalent to the preceding, but simplified */
A371344(n) = {my (Vmin=oo,CM, n2=n^2); forvec(v=vector(5,k,[1,n]), v[4]+v[5]= Vmin || Vmin=CM); Vmin/2} \\ M. F. Hasler, Dec 02 2024

a(33), a(37), a(38), and a(43) corrected by Hugo Pfoertner, Dec 03 2024

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

Original entry on oeis.org

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 120, 131, 144, 156, 168, 182, 196, 210, 225, 239, 256, 270, 288, 306, 321, 342, 361, 380, 399, 420, 441, 460, 484, 505, 527, 552, 576, 599, 623, 649, 673, 702, 729, 752, 781, 808, 840, 870, 900

Offset: 1

Hugo Pfoertner, Apr 16 2024

nonn

Cf. A316841, A371973, A371345.

See the formula section for the relationships with A002620, A173196, A316843, A316853.

A2(a,b,c) = {my(s=(a+b+c)/2);s*(s-a)*(s-b)*(s-c)};
a371972(n) = {my (A=List()); for(s2=1,n, for(s3=1,s2, if(s2+s3>n, listput(A, A2(n,s2,s3))))); #Set(A)};

a(n) <= A002620(n+1), with equality for n <= 20.

a(n) = |{A316853(k) : A316843(k) = n}| = |{A316853(k) : A173196(n) < k <= A173196(n+1)}|. - Peter Munn, Jul 30 2025

cp's OEIS Frontend

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

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A371070 a(n) is the number of distinct volumes > 0 of tetrahedra with the sum of their integer edge lengths equal to n.

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A371344 a(n)/144 is the minimum squared volume > 0 of a tetrahedron with integer edge lengths whose largest is n.

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A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

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