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%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
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