OFFSET
0,4
COMMENTS
The analog for 2 X 2 matrices turns out to be A000010(n), cf. mathoverflow post by user FFCH.
All terms > 3 are divisible by 9, and all a(2k) are even: this can be seen from symmetry arguments.
LINKS
Brendan McKay, Table of n, a(n) for n = 0..100
Pavel Gubkin, Number of matrices with unit determinant and fixed sum of elements, mathoverflow, Feb. 28, 2023
User FFCH, Decomposition of a natural number as sum of positive integers, mathoverflow, Feb. 23, 2023
FORMULA
a(n) / n^5 appears to have lim sup < 0.005 and lim inf > 0.003. [Observation by Brendan McKay, cf. Gubkin mathoverflow link.]
EXAMPLE
a(0) = a(1) = a(2) = 0, because a nonzero determinant isn't possible unless each of the 3 rows and columns have at least one nonzero entry.
a(3) = 3 counts the unit matrix and its two cyclic permutations M_ij = [i-j+-1 in 3Z].
PROG
(Python)
from sympy.utilities.iterables import multiset_permutations, partitions
def A361083(n):
c = 0
for s, d in partitions(n, m=9, size=True):
d.update({0:9-s})
c += sum(1 for p in multiset_permutations(d) if p[0]*(p[4]*p[8]-p[5]*p[7])-p[1]*(p[3]*p[8]-p[5]*p[6])+p[2]*(p[3]*p[7]-p[4]*p[6])==1)
return c # Chai Wah Wu, Mar 02 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Mar 01 2023
EXTENSIONS
Values a(15) and beyond from Brendan McKay, Mar 02 2023
STATUS
approved