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A280391 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant * n.

%I A280391 #22 May 22 2025 09:10:16
%S A280391 1,12,25,57,81,141,169,259,297,413,441,621,625,825,873,1079,1089,1403,
%T A280391 1369,1739,1729,2021,2025,2507,2433,2859,2905,3301,3249,4029,3721,
%U A280391 4509,4305,4793,4989,5551,5329,6027,6025,6807,6561,7917,7225,8357,8121,8677,8649,9843,9481,10889
%N A280391 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant * n.
%C A280391 All the values except a(1) are odd.
%C A280391 From _Robert Israel_, Jan 02 2017: (Start)
%C A280391 Number of solutions to (n+1)*x*y = (n-1)*z*w for x,y,z,w in [0..n].
%C A280391 a(n) >= (2n+1)^2, with equality if n+1 is an odd prime. (End)
%H A280391 Robert Israel and Indranil Ghosh, <a href="/A280391/b280391.txt">Table of n, a(n) for n = 0..1600</a> (n = 0..200 from Indranil Ghosh)
%p A280391 g:= proc(r,n) if r = 0 then 2*n+1 else nops(select(t -> t <= n and r <= t*n, numtheory:-divisors(r))) fi end proc:
%p A280391 f:= proc(n) local c;
%p A280391     if n::even then (2*n+1)^2 + add(g((n+1)*c,n)*g((n-1)*c,n), c=1..n-1)
%p A280391     else (2*n+1)^2 + add(g((n+1)/2*c,n) * g((n-1)/2*c,n), c=1..2*n-1)
%p A280391     fi
%p A280391 end proc:
%p A280391 map(f, [$0..100]); # _Robert Israel_, Jan 02 2017
%t A280391 g[r_, n_] := If[r == 0, 2n + 1, Length[Select[Divisors[r], # <= n && r <= # n&]]];
%t A280391 f[n_] := If[EvenQ[n], (2n + 1)^2 + Sum[g[(n + 1)c, n] g[(n - 1)c, n], {c, 1, n - 1}], (2n + 1)^2 + Sum[g[(n + 1)/2 c, n] g[(n - 1)/2 c, n], {c, 1, 2n - 1}]];
%t A280391 f /@ Range[0, 100] (* _Jean-François Alcover_, Jul 29 2020, after _Robert Israel_ *)
%o A280391 (Python)
%o A280391 def t(n):
%o A280391     s=0
%o A280391     for a in range(0,n+1):
%o A280391         for b in range(0,n+1):
%o A280391             for c in range(0,n+1):
%o A280391                 for d in range(0,n+1):
%o A280391                     if (a*d-b*c)*n==(a*d+b*c):
%o A280391                         s+=1
%o A280391     return s
%o A280391 for i in range(0,201):
%o A280391     print(t(i))
%Y A280391 Cf. A280321 (Number of 2 X 2 matrices with all elements in {0,..,n} with permanent*n = determinant).
%Y A280391 Cf. A015237 (Number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent).
%Y A280391 Cf. A016754 (Number of 2 X 2 matrices having all elements in {0..n} with determinant =2* permanent).
%Y A280391 Cf. A280364 (Number of 2 X 2 matrices having all elements in {0..n} with determinant^n = permanent).
%K A280391 nonn
%O A280391 0,2
%A A280391 _Indranil Ghosh_, Jan 02 2017