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Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.
1

%I #40 May 16 2020 03:26:36

%S 0,0,0,12,48,120,240,420,672,1008,1440,1980,2640,3432,4368,5460,6720,

%T 8160,9792,11628,13680,15960,18480,21252,24288,27600,31200,35100,

%U 39312,43848,48720,53940,59520,65472,71808,78540,85680,93240,101232,109668,118560

%N Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

%C Consider all Pythagorean triples (X,Y,Z=Y+2) ordered by increasing Z; A005843, A005563, A002522 and A007531 give the X, Y, Z and area A values of related triangles; for n >= 2 altitude h(n) = a(n+1)/A002522(n) or h(n)/2 is irreducible fraction in Q\Z. - _Ralf Steiner_, Mar 29 2020

%H Indranil Ghosh, <a href="/A280058/b280058.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.

%F a(n) == 12 (mod 12).

%F From _G. C. Greubel_, Dec 25 2016: (Start)

%F G.f.: (12*x^3)/(1 - x)^4.

%F E.g.f.: 2*x^3*exp(x).

%F a(n) = 2*n*(n-1)*(n-2).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

%F a(n) = 12 * A000292(n-2) for n>1. - _Alois P. Heinz_, Jan 30 2017

%F a(n+1) = sqrt(A016742(n)*A099761(n-1)) for n>=2. - _Ralf Steiner_, Mar 29 2020

%t Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* or *) LinearRecurrence[{4,-6, 4, -1}, {0, 0, 0, 12}, 50] (* _G. C. Greubel_, Dec 25 2016 *)

%o (Python)

%o def t(n):

%o s=0

%o for a in range(0,n+1):

%o for b in range(0,n+1):

%o if a!=b:

%o for c in range(0,n+1):

%o if a!=c and b!=c:

%o for d in range(0,n+1):

%o if d!=a and d!=b and d!=c:

%o if (a*d-b*c)==(a*d+b*c):

%o s+=1

%o return s

%o for i in range(0,201):

%o print str(i)+" "+str(t(i))

%o (PARI) for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ _G. C. Greubel_, Dec 25 2016

%o (PARI) a(n)=12*binomial(n,3) \\ _Charles R Greathouse IV_, Dec 25 2016

%Y Cf. A000292, A015237 (where the entries can be repeated), A005843, A005563, A002522, A016742, A099761, A007531.

%K nonn,easy

%O 0,4

%A _Indranil Ghosh_, Dec 25 2016