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A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)