A211034 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 1 (mod 3).

0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744, 10955, 15000, 18100, 23988, 31104, 36432, 46179, 57624, 66052, 81056, 98304, 110848, 132723, 157464, 175284, 205860, 240000, 264400, 305723, 351384, 383812, 438144, 497664, 539712, 609531

Offset

0,2

Comments

Also, the number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 2 (mod 3). A211033(n) + 2*A211034(n)=n^4 for n>0. For a guide to related sequences, see A210000.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).

Formula

From Chai Wah Wu, Nov 28 2016: (Start)

a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n > 12.

G.f.: -x*(4*x^9 + 20*x^8 + 59*x^7 + 109*x^6 + 96*x^5 + 136*x^4 + 100*x^3 + 28*x^2 + 21*x + 3)/((x - 1)^5*(x^2 + x + 1)^4).

If r = floor(n/3)+1, s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:

a(n) = r^2*s^2 + 2*r^2*s*t + r^2*t^2 + 2*r*s^3 + 6*r*s^2*t + 6*r*s*t^2 + 2*r*t^3 + 2*s^3*t + 2*s*t^3.

If n == 0 mod 3, then a(n) = 4*n^2*(2*n^2 + 6*n + 3)/27.

If n == 1 mod 3, then a(n) = (8*n^4 + 28*n^3 + 33*n^2 + 16*n - 4)/27.

If n == 2 mod 3, then a(n) = 8*(n + 1)^4/27. (End)

Mathematica

t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := u[n] = Sum[c[n, 3 k], {k, -2*n^2, 2*n^2}]
v[n_] := v[n] = Sum[c[n, 3 k + 1], {k, -2*n^2, 2*n^2}]
w[n_] := w[n] = Sum[c[n, 3 k + 2], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 0, z1}]   (* A211033 *)
Table[v[n], {n, 0, z1}]   (* A211034 *)
Table[w[n], {n, 0, z1}]   (* A211034 *)
LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744}, 60] (* Vincenzo Librandi, Nov 29 2016 *)

Prog

(Python)
from __future__ import division
def A211034(n):
    x, y, z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
    return x**2*y**2 + 2*x**2*y*z + x**2*z**2 + 2*x*y**3 + 6*x*y**2*z + 6*x*y*z**2 + 2*x*z**3 + 2*y**3*z + 2*y*z**3 # Chai Wah Wu, Nov 28 2016

Crossrefs

Cf. A210000, A211033.

Sequence in context: A363536 A293594 A160665 * A220868 A296273 A101008

Adjacent sequences: A211031 A211032 A211033 * A211035 A211036 A211037

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2012

Status

approved