A361083 Number of 3 X 3 matrices with unit determinant and nonnegative integer entries whose sum is n.

0, 0, 0, 3, 18, 54, 126, 261, 432, 783, 1134, 1899, 2286, 3960, 4680, 6876, 8262, 12654, 12618, 20799, 20934, 30024, 32760, 48141, 43632, 68976, 68094, 91161, 93042, 138006, 112194, 187227, 170982, 224892, 226728, 310824, 265770, 418410, 372384, 484920, 455400

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

Cf. A000010 (analog for 2 X 2 matrices).

Cf. A361082 (analog for 3 X 3 matrices with positive entries only).

Sequence in context: A238649 A268484 A085789 * A027334 A130505 A222204

Adjacent sequences: A361078 A361079 A361082 * A361085 A361086 A361088

Keyword

nonn

Author

M. F. Hasler, Mar 01 2023

Extensions

Values a(15) and beyond from Brendan McKay, Mar 02 2023

Status

approved