cp's OEIS Frontend

A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

%I A280058 #58 Jun 30 2025 04:28:31
%S A280058 0,0,0,12,48,120,240,420,672,1008,1440,1980,2640,3432,4368,5460,6720,
%T A280058 8160,9792,11628,13680,15960,18480,21252,24288,27600,31200,35100,
%U A280058 39312,43848,48720,53940,59520,65472,71808,78540,85680,93240,101232,109668,118560
%N A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.
%C A280058 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 A280058 Indranil Ghosh, <a href="/A280058/b280058.txt">Table of n, a(n) for n = 0..1000</a>
%H A280058 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A280058 a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.
%F A280058 a(n) = 2*n*(n-1)*(n-2). - _David Radcliffe_, Jun 14 2025
%F A280058 a(n) == 0 (mod 12).
%F A280058 From _G. C. Greubel_, Dec 25 2016: (Start)
%F A280058 G.f.: (12*x^3)/(1 - x)^4.
%F A280058 E.g.f.: 2*x^3*exp(x).
%F A280058 a(n) = 2*n*(n-1)*(n-2).
%F A280058 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%F A280058 a(n) = 12 * A000292(n-2) for n>1. - _Alois P. Heinz_, Jan 30 2017
%F A280058 a(n+1) = sqrt(A016742(n)*A099761(n-1)) for n>=2. - _Ralf Steiner_, Mar 29 2020
%F A280058 From _Amiram Eldar_, Jun 30 2025: (Start)
%F A280058 Sum_{n>=3} 1/a(n) = 1/8.
%F A280058 Sum_{n>=3} (-1)^(n+1)/a(n) = log(2) - 5/8. (End)
%t A280058 Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* _G. C. Greubel_, Dec 25 2016 *)
%o A280058 (Python)
%o A280058 def t(n):
%o A280058     s=0
%o A280058     for a in range(0,n+1):
%o A280058         for b in range(0,n+1):
%o A280058             if a!=b:
%o A280058                 for c in range(0,n+1):
%o A280058                     if a!=c and b!=c:
%o A280058                         for d in range(0,n+1):
%o A280058                             if d!=a and d!=b and d!=c:
%o A280058                                 if (a*d-b*c)==(a*d+b*c):
%o A280058                                     s+=1
%o A280058     return s
%o A280058 for i in range(0,201):
%o A280058     print(str(i)+" "+str(t(i)))
%o A280058 (Python) a = lambda n: 2*n*(n-1)*(n-2) # _David Radcliffe_, Jun 14 2025
%o A280058 (PARI) for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ _G. C. Greubel_, Dec 25 2016
%o A280058 (PARI) a(n)=12*binomial(n,3) \\ _Charles R Greathouse IV_, Dec 25 2016
%Y A280058 Cf. A000292, A015237 (where the entries can be repeated), A005843, A005563, A002522, A016742, A099761, A007531.
%K A280058 nonn,easy
%O A280058 0,4
%A A280058 _Indranil Ghosh_, Dec 25 2016