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

Author

Indranil Ghosh, Dec 12 2016

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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

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(f"{i} {t(i)}")

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)