A280058 Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.

0, 0, 0, 12, 48, 120, 240, 420, 672, 1008, 1440, 1980, 2640, 3432, 4368, 5460, 6720, 8160, 9792, 11628, 13680, 15960, 18480, 21252, 24288, 27600, 31200, 35100, 39312, 43848, 48720, 53940, 59520, 65472, 71808, 78540, 85680, 93240, 101232, 109668, 118560

Offset

0,4

Comments

Consider all Pythagorean triples (X,Y,Z=Y+2) ordered by increasing Z; A005843, A005563, A002522 and A007531 give the X, Y, Z and area A values of related triangles; for n >= 2 altitude h(n) = a(n+1)/A002522(n) or h(n)/2 is irreducible fraction in Q\Z. - Ralf Steiner, Mar 29 2020

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.

a(n) == 12 (mod 12).

From G. C. Greubel, Dec 25 2016: (Start)

G.f.: (12*x^3)/(1 - x)^4.

E.g.f.: 2*x^3*exp(x).

a(n) = 2*n*(n-1)*(n-2).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

a(n) = 12 * A000292(n-2) for n>1. - Alois P. Heinz, Jan 30 2017

a(n+1) = sqrt(A016742(n)*A099761(n-1)) for n>=2. - Ralf Steiner, Mar 29 2020

Mathematica

Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 0, 0, 12}, 50] (* G. C. Greubel, Dec 25 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-b*c)==(a*d+b*c):
                                    s+=1
    return s
for i in range(0, 201):
    print str(i)+" "+str(t(i))
(PARI) for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ G. C. Greubel, Dec 25 2016
(PARI) a(n)=12*binomial(n, 3) \\ Charles R Greathouse IV, Dec 25 2016

Crossrefs

Cf. A000292, A015237 (where the entries can be repeated), A005843, A005563, A002522, A016742, A099761, A007531.

Sequence in context: A256695 A135453 A165280 * A173548 A006564 A239352

Adjacent sequences: A280055 A280056 A280057 * A280059 A280060 A280061

Keyword

nonn,easy

Author

Indranil Ghosh, Dec 25 2016

Status

approved