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 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 28 22:21 EDT 2023. Contains 363028 sequences. (Running on oeis4.)