A280056 - OEIS

%I #31 Dec 26 2016 13:39:20

%S 0,0,0,8,48,144,360,720,1344,2240,3600,5400,7920,11088,15288,20384,

%T 26880,34560,44064,55080,68400,83600,101640,121968,145728,172224,

%U 202800,236600,275184,317520,365400,417600,476160,539648,610368,686664,771120,861840,961704,1068560,1185600

%N Number of 2 X 2 matrices with entries in {0,1,...,n} and even trace with no entries repeated.

%C a(n) mod 8 = 0.

%H Indranil Ghosh, <a href="/A280056/b280056.txt">Table of n, a(n) for n = 0..1000</a>

%H <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 a(n) = (1/4)*(n-3)*(n-2)*(2*n^2 - 4*n + 1 - (-1)^n) for n>=0.

%F From _Colin Barker_, Dec 25 2016: (Start)

%F a(n) = (n^4 - 3*n^3 + 2*n^2)/2 for n even.

%F a(n) = (n^4 - 3*n^3 + n^2 + 3*n - 2)/2 for n odd.

%F 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 G.f.: 8*x^3*(1 + 3*x)*(1 + x + x^2) / ((1 - x )^5*(1 + x)^3).

%F (End)

%F These formulas are true. a(n) = ((-1)^n + 2*n^2 - 1)*(n-1)*(n-2)/4 = (n^2 - p(n))*C(n-1,2), where p(n) is the parity of n, i.e., p(n) = 0 if n is even and p(n) = 1 if n is odd. - _Chai Wah Wu_, Dec 25 2016

%F E.g.f.: (1/4)*((-6 -4*x - x^2)*exp(-x) + (6- 8*x + 5*x^2 + 2*x^3 + 2*x^4 )*exp(x)). - _G. C. Greubel_, Dec 26 2016

%t Table[(1/4)*(n - 3)*(n - 2)*(2*n^2 - 4*n + 1 - (-1)^n), {n, 0, 50}] (* _G. C. Greubel_, Dec 26 2016 *)

%o (Python)

%o def a(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)%2==0:

%o                                     s+=1

%o     return s

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

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

%o (Python)

%o def A280056(x):

%o     return -((x**2-5*x+6)*(-2*x**2+4*x+(-1)**x-1))/4

%o (Python)

%o from __future__ import division

%o def A280056(n):

%o     return (n**2 - (n % 2))*(n-1)*(n-2)//2 # _Chai Wah Wu_, Dec 25 2016

%o (PARI) concat(vector(3), Vec(8*x^3*(1 + 3*x)*(1 + x + x^2) / ((1 - x )^5*(1 + x)^3) + O(x^30))) \\ _Colin Barker_, Dec 25 2016

%Y Cf. A210378 (where the elements can be repeated).

%K nonn,easy

%O 0,4

%A _Indranil Ghosh_, Dec 24 2016