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A211071 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 1 (mod 3). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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: A209456 A069145 A264184 * A212135 A210569 A334581

Adjacent sequences:  A211068 A211069 A211070 * A211072 A211073 A211074

KEYWORD

nonn

AUTHOR

Clark Kimberling, Apr 01 2012

STATUS

approved

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Last modified January 27 11:16 EST 2022. Contains 350607 sequences. (Running on oeis4.)