A371071 -id:A371071 - cp's OEIS frontend

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

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

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};

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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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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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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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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