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A280344 Number of 2 X 2 matrices with all elements in {0,...,n} with determinant = permanent^n.

%I A280344 #45 Aug 08 2025 20:23:12
%S A280344 0,12,30,56,90,132,182,240,306,380,462,552,650,756,870,992,1122,1260,
%T A280344 1406,1560,1722,1892,2070,2256,2450,2652,2862,3080,3306,3540,3782,
%U A280344 4032,4290,4556,4830,5112,5402,5700,6006,6320,6642,6972,7310,7656,8010,8372,8742,9120,9506,9900,10302
%N A280344 Number of 2 X 2 matrices with all elements in {0,...,n} with determinant = permanent^n.
%C A280344 For n>0, a(n) is the perimeter of a primitive Pythagorean triangle. - _Torlach Rush_, Jul 11 2019
%H A280344 Indranil Ghosh, <a href="/A280344/b280344.txt">Table of n, a(n) for n = 0..995</a>
%F A280344 a(0) = A002939(0) = 0; a(n) = A002939(n+1), for n>=1.
%F A280344 a(n) = (((n-2)*a(n-1))/(n-4)) - (6*(3*(n-1)+1)/(n-4)) for n>=4.
%F A280344 Conjectures from _Colin Barker_, Jan 01 2017: (Start)
%F A280344 a(n) = 2 + 6*n + 4*n^2 for n>0.
%F A280344 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
%F A280344 G.f.: 2*x*(6 - 3*x + x^2) / (1 - x)^3.
%F A280344 (End)
%F A280344 From _Torlach Rush_, Jul 11 2019: (Start)
%F A280344 a(n) = (2*n+1)*(2*n+2), n>0.
%F A280344 a(n) = 2*((n+1)^2 + ((n+1)*n)), n>0.
%F A280344 (End)
%t A280344 Table[Boole[n != 0] 2 # (2 # - 1) &[n + 1], {n, 0, 50}] (* or *)
%t A280344 CoefficientList[Series[2 x (6 - 3 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Michael De Vlieger_, Jan 01 2017 *)
%o A280344 (Python)
%o A280344 def t(n):
%o A280344     s=0
%o A280344     for a in range(0,n+1):
%o A280344         for b in range(0,n+1):
%o A280344             for c in range(0,n+1):
%o A280344                 for d in range(0,n+1):
%o A280344                     if (a*d-b*c)==(a*d+b*c)**n:
%o A280344                         s+=1
%o A280344     return s
%o A280344 for i in range(0,41):
%o A280344     print(i, t(i))
%Y A280344 Same as both A002939 and A118239 without A002939(1) = 2.
%Y A280344 Cf. A016754, A280343.
%K A280344 nonn
%O A280344 0,2
%A A280344 _Indranil Ghosh_, Jan 01 2017