A361083 - OEIS

%I #36 Mar 11 2023 06:22:16

%S 0,0,0,3,18,54,126,261,432,783,1134,1899,2286,3960,4680,6876,8262,

%T 12654,12618,20799,20934,30024,32760,48141,43632,68976,68094,91161,

%U 93042,138006,112194,187227,170982,224892,226728,310824,265770,418410,372384,484920,455400

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

%C The analog for 2 X 2 matrices turns out to be A000010(n), cf. mathoverflow post by user FFCH.

%C All terms > 3 are divisible by 9, and all a(2k) are even: this can be seen from symmetry arguments.

%H Brendan McKay, <a href="/A361083/b361083.txt">Table of n, a(n) for n = 0..100</a>

%H Pavel Gubkin, <a href="https://mathoverflow.net/questions/441828/">Number of matrices with unit determinant and fixed sum of elements</a>, mathoverflow, Feb. 28, 2023

%H User FFCH, <a href="https://mathoverflow.net/questions/441469/">Decomposition of a natural number as sum of positive integers</a>, mathoverflow, Feb. 23, 2023

%F a(n) / n^5 appears to have lim sup < 0.005 and lim inf > 0.003. [Observation by Brendan McKay, cf. Gubkin mathoverflow link.]

%e 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.

%e a(3) = 3 counts the unit matrix and its two cyclic permutations M_ij = [i-j+-1 in 3Z].

%o (Python)

%o from sympy.utilities.iterables import multiset_permutations, partitions

%o def A361083(n):

%o     c = 0

%o     for s,d in partitions(n,m=9,size=True):

%o         d.update({0:9-s})

%o         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)

%o     return c # _Chai Wah Wu_, Mar 02 2023

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

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

%K nonn

%O 0,4

%A _M. F. Hasler_, Mar 01 2023

%E Values a(15) and beyond from _Brendan McKay_, Mar 02 2023