A280588 - OEIS

%I #26 Jun 06 2020 13:01:11

%S 1,1,2,9,18,41,58,97,130,185,226,313,354,457,538,649,738,889,954,1145,

%T 1266,1449,1578,1809,1930,2177,2362,2609,2770,3129,3242,3609,3810,

%U 4097,4402,4793,5026,5433,5674,6097,6346,6929,7090,7641,8010,8433,8810,9369,9626,10297,10690

%N Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = determinant.

%H Indranil Ghosh and Chai Wah Wu, <a href="/A280588/b280588.txt">Table of n, a(n) for n = 0..10000</a> (terms for n = 0..200 from Indranil Ghosh)

%e For n = 4, the possible matrices are [0,0,0,0], [2,0,0,2], [2,0,1,3],[2,0,2,4], [2,1,0,3], [2,2,0,4], [3,0,1,2], [3,0,3,3], [3,1,0,2], [3,1,1,3], [3,1,2,4], [3,2,1,4], [3,3,0,3], [4,0,2,2], [4,1,2,3],

%e [4,2,0,2], [4,2,1,3] and [4,2,2,4]. There are 18 possibilities.

%e Here each of the matrices are defined as M = [a,b,c,d], where a = M[1][1], b = M[1][2], c = M[2][1] and d = M[2][2].

%e So, for n = 4, a(n) = 18.

%o (Python)

%o def t(n):

%o     s=0

%o     for a in range(n+1):

%o         for b in range(n+1):

%o             for c in range(n+1):

%o                 for d in range(n+1):

%o                     if (a+b+c+d)==(a*d-b*c):

%o                         s+=1

%o     return s

%o for i in range(51):

%o     print(str(i)+" "+str(t(i)))

%Y Cf. A210374 (Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n+2).

%K nonn

%O 0,3

%A _Indranil Ghosh_, Jan 10 2017