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

0, 0, 0, 16, 72, 216, 480, 960, 1680, 2800, 4320, 6480, 9240, 12936, 17472, 23296, 30240, 38880, 48960, 61200, 75240, 91960, 110880, 133056, 157872, 186576, 218400, 254800, 294840, 340200, 389760, 445440, 505920, 573376, 646272, 727056, 813960, 909720

Offset: 0

Author

Indranil Ghosh, Dec 26 2016

Keywords

• nonn
• easy

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A210379 (where all elements can be repeated).

Programs

Mathematica

LinearRecurrence[{2,2,-6,0,6,-2,-2,1}, {0,0,0,16,72,216,480,960}, 50] (* G. C. Greubel, Dec 26 2016 *)

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

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

Python

def a(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            if a!=b:
                for c in range(0,n+1):
                    if a!=c and b!=c:
                        for d in range(0,n+1):
                            if d!=a and d!=b and d!=c:
                                if (a+d)%2==1:
                                    s+=1
    return s
print([a(n) for n in range(41)])

Python

def a(n):
    return ((n-3)*(n-2)*(2*n**2+(-1)**n-1))//4

Formula

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

From Colin Barker, Dec 26 2016: (Start)

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

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

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.

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

(End)

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