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A262719 a(n) is the smallest nonnegative k such that there is no 3 X 3 matrix with entries in {1,...,n} whose determinant is k.

%I A262719 #20 Nov 14 2024 09:51:15
%S A262719 1,6,21,55,110,203,357,544,808,1177,1670,2215,2865,3599,4558,5621,
%T A262719 6637,8041,9769,11413,13394,15593,17683,20317,23249,26063,29506,33287,
%U A262719 37461,41692,46306,50707,55667,61723,67547,73939,80767,87941,94913,101613,111422
%N A262719 a(n) is the smallest nonnegative k such that there is no 3 X 3 matrix with entries in {1,...,n} whose determinant is k.
%H A262719 Hiroaki Yamanouchi, <a href="/A262719/b262719.txt">Table of n, a(n) for n = 1..50</a>
%e A262719 For n=1, the only matrix is the matrix of all 1s, which has determinant 0. Hence, a(1)=1.
%o A262719 (Python)
%o A262719 from itertools import product, groupby, count
%o A262719 def det(m):
%o A262719     a, b, c, d, e, f, g, h, i = m
%o A262719     return abs(a*(e*i-f*h)-b*(d*i-f*g)+c*(d*h-e*g))
%o A262719 def a262719(n):
%o A262719     s = list(product(range(1, n+1), repeat=9))
%o A262719     i = 0
%o A262719     for k, ms in groupby(sorted(s, key=det), key=det):
%o A262719         if k!=i:
%o A262719             return i
%o A262719         i += 1
%o A262719     return i
%K A262719 nonn
%O A262719 1,2
%A A262719 _Christian Perfect_, Sep 28 2015
%E A262719 a(7)-a(41) from _Hiroaki Yamanouchi_, Oct 17 2015