cp's OEIS Frontend

A279905 Number of 2 X 2 matrices with entries in {0,1,...,n} and odd trace with no elements repeated.

%I A279905 #33 Aug 18 2025 18:48:13
%S A279905 0,0,0,16,72,216,480,960,1680,2800,4320,6480,9240,12936,17472,23296,
%T A279905 30240,38880,48960,61200,75240,91960,110880,133056,157872,186576,
%U A279905 218400,254800,294840,340200,389760,445440,505920,573376,646272,727056,813960,909720
%N A279905 Number of 2 X 2 matrices with entries in {0,1,...,n} and odd trace with no elements repeated.
%H A279905 Indranil Ghosh, <a href="/A279905/b279905.txt">Table of n, a(n) for n = 0..1000</a>
%H A279905 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A279905 a(n) = ((n-2)*(n-1)*(2*(n+1)^2-(-1)^n-1))/4 for n>=0 .
%F A279905 From _Colin Barker_, Dec 26 2016: (Start)
%F A279905 a(n) = (n^4 - n^3 - 4*n^2 + 4*n)/2 for n even.
%F A279905 a(n) = (n^4 - n^3 - 3*n^2 + n + 2)/2 for n odd.
%F A279905 a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>7.
%F A279905 G.f.: 8*x^3*(2 + 5*x + 5*x^2) / ((1 - x)^5*(1 + x)^3).
%F A279905 (End)
%F A279905 E.g.f.: (1/4)*((-2 - 2*x - x^2)*exp(-x) + (2 -2*x + x^2 + 10*x^3 + 2*x^4 )*exp(x)). - _G. C. Greubel_, Dec 26 2016
%t A279905 LinearRecurrence[{2,2,-6,0,6,-2,-2,1}, {0,0,0,16,72,216,480,960}, 50] (* _G. C. Greubel_, Dec 26 2016 *)
%o A279905 (Python)
%o A279905 def a(n):
%o A279905     s=0
%o A279905     for a in range(0,n+1):
%o A279905         for b in range(0,n+1):
%o A279905             if a!=b:
%o A279905                 for c in range(0,n+1):
%o A279905                     if a!=c and b!=c:
%o A279905                         for d in range(0,n+1):
%o A279905                             if d!=a and d!=b and d!=c:
%o A279905                                 if (a+d)%2==1:
%o A279905                                     s+=1
%o A279905     return s
%o A279905 print([a(n) for n in range(41)])
%o A279905 (Python)
%o A279905 def a(n):
%o A279905     return ((n-3)*(n-2)*(2*n**2+(-1)**n-1))//4
%o A279905 (PARI) concat(vector(3), Vec(8*x^3*(2 + 5*x + 5*x^2) / ((1 - x)^5*(1 + x)^3) + O(x^50))) \\ _Colin Barker_, Dec 26 2016
%o A279905 (PARI) concat([0,0,0], Vec(8*x^3*(2 + 5*x + 5*x^2) / ((1 - x)^5*(1 + x)^3) + O(x^50))) \\ _G. C. Greubel_, Dec 26 2016
%Y A279905 Cf. A210379 (where all elements can be repeated).
%K A279905 nonn,easy
%O A279905 0,4
%A A279905 _Indranil Ghosh_, Dec 26 2016