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

0,4

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).

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

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 *)

Prog

(Python)
def t(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
for i in range(0, 201):
    print str(i)+" "+str(t(i))
(Python)
def a(x):
    return ((x-3)*(x-2)*(2*x**2+(-1)**x-1))/4
(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

Crossrefs

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

Sequence in context: A339993 A232402 A212513 * A146748 A029872 A056633

Adjacent sequences: A279902 A279903 A279904 * A279906 A279907 A279908

Keyword

nonn,easy

Author

Indranil Ghosh, Dec 26 2016

Status

approved