A211071 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 1 (mod 3).

0, 4, 24, 83, 208, 384, 756, 1332, 1944, 3099, 4672, 6144, 8768, 12100, 15000, 19995, 26064, 31104, 39588, 49588, 57624, 70931, 86272, 98304, 117984, 140292, 157464, 185283, 216400, 240000, 277940, 319924, 351384, 401643, 456768

Offset

1,2

Comments

Also, the number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 2 (mod 3).

A210698(n) + 2*A211071(n) = n^4.

For a guide to related sequences, see A210000.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..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 30 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 > 13.

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

If r = floor(n/3), 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) = 8*n^4/27.

If n == 1 mod 3, then a(n) = (8*n^4 + 4*n^3 - 3*n^2 - 2*n - 7)/27.

If n == 2 mod 3, then a(n) = (8*n^4 + 8*n^3 - 12*n^2 - 16*n - 4)/27. (End)

Mathematica

a = 1; b = n; z1 = 45;
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, 1, z1}] (* A210698 *)
Table[v[n], {n, 1, z1}] (* A211071 *)
Table[w[n], {n, 1, z1}] (* A211071 *)
LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 4, 24, 83, 208, 384, 756, 1332, 1944, 3099, 4672, 6144, 8768}, 40] (* Vincenzo Librandi, Dec 01 2016 *)

Prog

(Python)
from __future__ import division
def A211071(n):
    if n % 3 == 0:
        return 8*n**4//27
    elif n % 3 == 1:
        return (8*n**4 + 4*n**3 - 3*n**2 - 2*n - 7)//27
    else:
        return (8*n**4 + 8*n**3 - 12*n**2 - 16*n - 4)//27 # Chai Wah Wu, Nov 30 2016

Crossrefs

Cf. A210000, A210698, A211034.

Sequence in context: A069145 A264184 A354476 * A212135 A210569 A334581

Adjacent sequences: A211068 A211069 A211070 * A211072 A211073 A211074

Keyword

nonn

Author

Clark Kimberling, Apr 01 2012

Status

approved