A211033 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 0 (mod 3).

1, 10, 33, 152, 297, 528, 1217, 1834, 2673, 4744, 6385, 8448, 13073, 16506, 20625, 29336, 35545, 42768, 57457, 67642, 79233, 102152, 117729, 135168, 168929, 191530, 216513, 264088, 295561, 330000, 394721, 437130, 483153, 568712, 624337, 684288, 794737, 866074

Offset

0,2

Comments

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^11 - 7*x^10 - 25*x^9 - 53*x^8 - 91*x^7 - 219*x^6 - 139*x^5 - 109*x^4 - 115*x^3 - 23*x^2 - 9*x - 1)/((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^4 + 4*r^3*s + 4*r^3*t + 4*r^2*s^2 + 8*r^2*s*t + 4*r^2*t^2 + s^4 + 6*s^2*t^2 + t^4.

If n == 0 mod 3, then a(n) = (11*n^4 + 60*n^3 + 138*n^2 + 108*n)/27 + 1.

If n == 1 mod 3, then a(n) = (11*n^4 + 52*n^3 + 96*n^2 + 76*n + 35)/27.

If n == 2 mod 3, then a(n) = 11*(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 *)

Prog

(Python)
from __future__ import division
def A211033(n):
    x, y, z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
    return x**4 + 4*x**3*y + 4*x**3*z + 4*x**2*y**2 + 8*x**2*y*z + 4*x**2*z**2 + y**4 + 6*y**2*z**2 + z**4 # Chai Wah Wu, Nov 28 2016

Crossrefs

Cf. A210000, A211034.

Sequence in context: A094170 A373129 A004638 * A020479 A219818 A264251

Adjacent sequences: A211030 A211031 A211032 * A211034 A211035 A211036

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2012

Status

approved