A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated.

0, 0, 0, 16, 96, 216, 600, 1008, 2064, 3040, 5280, 7200, 11280, 14616, 21336, 26656, 36960, 44928, 59904, 71280, 92160, 107800, 135960, 156816, 193776, 220896, 268320, 302848, 362544, 405720, 479640, 532800, 623040, 687616, 796416, 873936, 1003680, 1095768, 1248984, 1357360, 1536720, 1663200

Offset

0,4

Comments

a(n) mod 8 = 0.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Formula

From Colin Barker and Charles R Greathouse IV, Dec 12 2016: (Start)

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>8.

a(n) = (5*n^4 - 8*n^3 + 4*n^2 - 16*n)/8 for n even.

a(n) = (5*n^4 - 12*n^3 + 2*n^2 + 12*n - 7)/8 for n odd.

G.f.: 8*x^3*(2 + 10*x + 7*x^2 + 8*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^4).

(End)

Prog

(Python)
def t(n):
    s=0
    for a in range(0, n+1):
        for b in range(0, n+1):
            for c in range(0, n+1):
                for d in range(0, n+1):
                   if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
                        if (a*d-b*c)%2==0:
                            s+=1
    return s
for i in range(0, 201):
    print str(i)+" "+str(t(i))
(PARI) F(n, {r=0})={my(s=vector(2), v); forvec(y=vector(4, j, [0, n]), for(k=23*!!r, 23, v=numtoperm(4, k); s[1+(y[v[1]]*y[v[4]]-y[v[3]]*y[v[2]])%2]++), 2*!r); return(s)} \\ Use r=1 for A210369;
a(n)=F(n, 0)[1]; \\ Also works for A210370 if F(n, 1)[2] is used instead. - R. J. Cano, Dec 12 2016
(PARI) a(n)=my(e=n\2+1, o=(n+1)\2); 24*binomial(o, 4) + 16*binomial(e, 2)*binomial(o, 2) + 24*o*binomial(e, 3) + 24*binomial(e, 4) \\ Charles R Greathouse IV, Dec 12 2016

Crossrefs

Cf. A210369 (where the entries can be repeated).

Sequence in context: A321338 A128702 A322639 * A192037 A143060 A006637

Adjacent sequences: A277041 A277042 A277043 * A277045 A277046 A277047

Keyword

nonn,easy

Author

Indranil Ghosh, Dec 12 2016

Status

approved