A210698 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 0 (mod 3).

1, 8, 33, 90, 209, 528, 889, 1432, 2673, 3802, 5297, 8448, 11025, 14216, 20625, 25546, 31393, 42768, 51145, 60824, 79233, 92394, 107297, 135168, 154657, 176392, 216513, 244090, 274481, 330000, 367641, 408728, 483153, 533050, 587089

Offset: 1

Author

Clark Kimberling, Apr 01 2012

Keywords

• nonn

Comments

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).

Crossrefs

Cf. A210000, A211071, A211033.

Programs

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}, {1, 8, 33, 90, 209, 528, 889, 1432, 2673, 3802, 5297, 8448, 11025}, 40] (* Vincenzo Librandi, Dec 01 2016 *)

Python

from _future_ import division
def A210698(n):
    if n % 3 == 0:
        return 11*n**4//27
    elif n % 3 == 1:
        return (11*n**4 - 8*n**3 + 6*n**2 + 4*n + 14)//27
    else:
        return (11*n**4 - 16*n**3 + 24*n**2 + 32*n + 8)//27 # Chai Wah Wu, Nov 30 2016

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*(-x^11 - 9*x^10 - 23*x^9 - 115*x^8 - 109*x^7 - 139*x^6 - 219*x^5 - 91*x^4 - 53*x^3 - 25*x^2 - 7*x - 1)/((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^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^4/27.

If n == 1 mod 3, then a(n) = (11*n^4 - 8*n^3 + 6*n^2 + 4*n + 14)/27.

If n == 2 mod 3, then a(n) = (11*n^4 - 16*n^3 + 24*n^2 + 32*n + 8)/27. (End)