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A280321 Number of 2 X 2 matrices with all elements in {0,..,n} having determinant = n*permanent.

%I A280321 #27 Jul 04 2023 12:35:22
%S A280321 1,12,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089,1225,
%T A280321 1369,1521,1681,1849,2025,2209,2401,2601,2809,3025,3249,3481,3721,
%U A280321 3969,4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569,7921,8281,8649,9025,9409,9801,10201
%N A280321 Number of 2 X 2 matrices with all elements in {0,..,n} having determinant = n*permanent.
%C A280321 Same as A016754, except for n=1. Here a(1)=12 but A016754(1)=9.
%H A280321 Indranil Ghosh, <a href="/A280321/b280321.txt">Table of n, a(n) for n = 0..990</a>
%F A280321 a(n+1) = (((n-2)*a(n))/(n-1)) + ((12*(n)^2-12*(n)+1)/(n-1)) for n>=1.
%F A280321 Conjectures from _Colin Barker_, Jan 01 2017: (Start)
%F A280321 a(n) = (1 + 2*n)^2 = A273789(n) = A273743(n) for n>1.
%F A280321 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
%F A280321 G.f.: (1 + 9*x - 8*x^2 + 9*x^3 - 3*x^4) / (1 - x)^3.
%F A280321 (End)
%o A280321 (Python)
%o A280321 def t(n):
%o A280321     s=0
%o A280321     for a in range(n+1):
%o A280321         for b in range(n+1):
%o A280321             for c in range(n+1):
%o A280321                 for d in range(n+1):
%o A280321                     if (a*d-b*c)==n*(a*d+b*c):
%o A280321                         s+=1
%o A280321     return s
%o A280321 for i in range(41):
%o A280321     print(str(i)+" "+str(t(i)))
%Y A280321 Cf. A016754.
%K A280321 nonn
%O A280321 0,2
%A A280321 _Indranil Ghosh_, Jan 01 2017