A371070 - OEIS

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

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OFFSET

6,7

LINKS

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?

FORMULA

a(n) <= A208454(n).

PROG

(PARI) a371070(n) = {my (L=List()); forpart (w=n, 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); if(v[1]+v[2]<v[4], next); if(v[1]+v[4]<v[2], next); if(v[2]+v[4]<v[1], next); if(v[1]+v[3]<v[5], next); if(v[1]+v[5]<v[3], next); if(v[3]+v[5]<v[1], next); if(v[2]+v[3]<v[6], next); if(v[2]+v[6]<v[3], next); if(v[3]+v[6]<v[2], next); my(CM=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])); if (CM>0, listput (L, CM))), [1, n], [6, 6]); #Set(Vec(L))};

(Python)

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+y<z or x+z<y or y+z<x)

TR = lambda x, y, z, t, u, v: TR1(t, u, v) and TR1(x, y, t) and TR1(x, z, u) and TR1(y, z, v)

c, d, sq = 0, set(), tuple(x**2 for x in range(n+1))

for s, w in partitions(n, m=6, k=n-5, size=True):

if s == 6:

for v in multiset_permutations(Counter(w).elements()):

if TR(*v) and (M:=CM(sq[v[0]], sq[v[1]], sq[v[2]], sq[v[3]], sq[v[4]], sq[v[5]]))>0 and M not in d: