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

1,2

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A210000, A211071, A211033.

Sequence in context: A140867 A212133 A212574 * A114105 A316148 A014820

Adjacent sequences: A210695 A210696 A210697 * A210699 A210700 A210701

Keyword

nonn

Author

Clark Kimberling, Apr 01 2012

Status

approved