cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A280844 Number of 2 X 2 matrices with entries in {-n,..,0,..,n} with no entries repeated having permanent = trace^n.

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%I A280844 #13 Jun 08 2025 04:58:16
%S A280844 0,0,0,4,16,20,16,36,56,60,72,76,80,100,112,100,136,124,152,172,192,
%T A280844 196,224,196,232,236,264,252,288,276,288,308,344,332,344,332,384,388,
%U A280844 416,404,456,428,456,444,496,468,512,468,536,556,648
%N A280844 Number of 2 X 2 matrices with entries in {-n,..,0,..,n} with no entries repeated having permanent = trace^n.
%C A280844 a(n) is also equal to the number of 2 X 2 matrices with entries in {-n,..,0,..n} with no elements repeated having determinant = trace^n except a(4). For permanent = trace^n, a(4) = 16 but for determinant = trace^n, a(4) = 24.
%C A280844 a(n) mod 4 = 0.
%C A280844 All solutions have trace -1, 0, or 1. The number of solutions with trace 0 is 8 * A132345(n), and the number of solutions with trace -1 is equal to the number of solutions with trace 1 when n is even. - _David Radcliffe_, Jun 08 2025
%H A280844 David Radcliffe, <a href="/A280844/b280844.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..103 from Indranil Ghosh).
%e A280844 For n = 5, the possible matrices are [-3,-5,-1,2], [-3,-1,-5,2],
%e A280844 [-3,1,5,2], [-3,5,1,2], [-2,-4,-1,2], [-2,-1,-4,2], [-2,1,4,2], [-2,4,1,2], [-1,1,3,2], [-1,3,1,2], [2,-5,-1,-3], [2,-4,-1,-2],
%e A280844 [2,-1,-5,-3], [2,-1,-4,-2], [2,1,3,-1], [2,1,4,-2], [2,1,5,-3],
%e A280844 [2,3,1,-1], [2,4,1,-2] and [2,5,1,-3].
%e A280844 Here each of the matrices 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 20 possibilities. So, for n = 5, a(n) = 20.
%o A280844 (Python)
%o A280844 def t(n):
%o A280844     s=0
%o A280844     for a in range(-n,n+1):
%o A280844         for b in range(-n,n+1):
%o A280844             if a!=b:
%o A280844                 for c in range(-n,n+1):
%o A280844                     if a!=c and b!=c:
%o A280844                         for d in range(-n,n+1):
%o A280844                             if d!=a and d!=b and d!=c:
%o A280844                                 if (a*d+b*c)==(a+d)**n:
%o A280844                                     s+=1
%o A280844     return s
%o A280844 for i in range(0,24):
%o A280844     print(t(i), end=', ')
%Y A280844 Cf. A278933, A278902, A279018, A132345
%K A280844 nonn
%O A280844 0,4
%A A280844 _Indranil Ghosh_, Jan 09 2017