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A280364 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant^n.

%I A280364 #44 Aug 14 2025 09:13:33
%S A280364 0,12,35,56,99,132,195,240,323,380,483,552,675,756,899,992,1155,1260,
%T A280364 1443,1560,1763,1892,2115,2256,2499,2652,2915,3080,3363,3540,3843,
%U A280364 4032,4355,4556,4899,5112,5475,5700,6083,6320,6723,6972,7395,7656,8099,8372,8835,9120,9603,9900,10403
%N A280364 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant^n.
%H A280364 Indranil Ghosh, <a href="/A280364/b280364.txt">Table of n, a(n) for n = 0..995</a>
%F A280364 a(n) = (((-2*n-1)*a(n-1))/(2*n-1)) + 8*n^2 + 10*n + 3 for n>=2. [Corrected by _David Radcliffe_, Aug 13 2025]
%F A280364 Conjectures from _Colin Barker_, Jan 01 2017: (Start)
%F A280364 a(n) = 4*n^2 + 8*n + 3 for n>0 and even.
%F A280364 a(n) = 4*n^2 + 6*n + 2 for n odd.
%F A280364 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
%F A280364 G.f.: x*(12 + 23*x - 3*x^2 - 3*x^3 + 3*x^4) / ((1 - x)^3*(1 + x)^2). (End)
%e A280364 For n=2, the matrices are [0,0,0,0], [0,0,0,1], [0,0,0,2], [0,0,1,0], [0,0,1,1], [0,0,1,2], [0,0,2,0], [0,0,2,1], [0,0,2,2], [0,1,0,0], [0,1,0,1], [0,1,0,2], [0,1,1,0], [0,1,1,1], [0,1,1,2], [0,2,0,0], [0,2,0,1], [0,2,0,2], [1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,0,2,0], [1,0,2,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,2,0,0], [1,2,0,1], [2,0,0,0], [2,0,1,0], [2,0,2,0], [2,1,0,0], [2,1,1,0], [2,2,0,0].
%e A280364 Here each of these matrices M is defined as  M=[a,b,c,d], where a=M[1][1], b=M[1][2], c=M[2][1], d=M[2][2]. There are 35 possibilities. So for n=2, a(n)=35.
%t A280364 CoefficientList[Series[x (12 + 23 x - 3 x^2 - 3 x^3 + 3 x^4)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* _Michael De Vlieger_, Jan 01 2017 *)
%o A280364 (Python)
%o A280364 def t(n):
%o A280364     s=0
%o A280364     for a in range(0,n+1):
%o A280364         for b in range(0,n+1):
%o A280364             for c in range(0,n+1):
%o A280364                 for d in range(0,n+1):
%o A280364                     if (a*d-b*c)**n==(a*d+b*c):
%o A280364                         s+=1
%o A280364     return s
%o A280364 for i in range(0,51):
%o A280364     print(i, t(i))
%o A280364 (Python)
%o A280364 def a(n):
%o A280364     if n==0:
%o A280364         return 0
%o A280364     if n==1:
%o A280364         return 12
%o A280364     return (((-2*n-1)*a(n-1))//(2*n-1))+8*n**2+10*n+3
%o A280364 for i in range(0,51):
%o A280364     print(i, a(i))
%Y A280364 Cf. A280344 (Number of 2 X 2 matrices with all elements in {0,...,n} with determinant = permanent^n).
%K A280364 nonn
%O A280364 0,2
%A A280364 _Indranil Ghosh_, Jan 01 2017