

A361083


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


2



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
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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



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 = [ij+1 in 3Z].


PROG

(Python)
from sympy.utilities.iterables import multiset_permutations, partitions
c = 0
for s, d in partitions(n, m=9, size=True):
d.update({0:9s})
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)


CROSSREFS

Cf. A000010 (analog for 2 X 2 matrices).
Cf. A361082 (analog for 3 X 3 matrices with positive entries only).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



