A279483 Number of 2 X 2 matrices with entries in {0,1,...,n} and odd determinant with no entry repeated.

0, 0, 0, 8, 24, 144, 240, 672, 960, 2000, 2640, 4680, 5880, 9408, 11424, 17024, 20160, 28512, 33120, 45000, 51480, 67760, 76560, 98208, 109824, 137904, 152880, 188552, 207480, 252000, 275520, 330240, 359040, 425408, 460224, 539784, 581400, 675792, 725040, 836000, 893760, 1023120, 1090320, 1240008

Offset: 0

Author

Indranil Ghosh, Dec 13 2016

Keywords

• nonn
• easy

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. A210370 (where the entries can be repeated).

Programs

Mathematica

CoefficientList[Series[8 x^3*(1 + 2 x + 11 x^2 + 4 x^3)/((1 - x)^5*(1 + x)^4), {x, 0, 43}], x] (* Michael De Vlieger, Dec 13 2016 *)

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)} \\ a(n)=F(n, 0)[2];

PARI

concat(vector(3), Vec(8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Dec 13 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==1:
                            s+=1
    return s
for i in range(0,201):
    print(i, t(i))

Formula

From Colin Barker, Dec 13 2016: (Start)

a(n) = (3*n^4 - 8*n^3 - 12*n^2 + 32*n)/8 for n even.

a(n) = (3*n^4 - 4*n^3 - 10*n^2 + 4*n + 7)/8 for n odd.

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.

G.f.: 8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4).

(End)