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

%I #17 Mar 03 2024 18:42:41

%S 0,4,24,83,208,384,756,1332,1944,3099,4672,6144,8768,12100,15000,

%T 19995,26064,31104,39588,49588,57624,70931,86272,98304,117984,140292,

%U 157464,185283,216400,240000,277940,319924,351384,401643,456768

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

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

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

%C For a guide to related sequences, see A210000.

%H Chai Wah Wu, <a href="/A211071/b211071.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1).

%F From _Chai Wah Wu_, Nov 30 2016: (Start)

%F 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.

%F 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).

%F If r = floor(n/3), s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:

%F 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.

%F If n == 0 mod 3, then a(n) = 8*n^4/27.

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

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

%t a = 1; b = n; z1 = 45;

%t t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]

%t c[n_, k_] := c[n, k] = Count[t[n], k]

%t u[n_] := u[n] = Sum[c[n, 3 k], {k, -2*n^2, 2*n^2}]

%t v[n_] := v[n] = Sum[c[n, 3 k + 1], {k, -2*n^2, 2*n^2}]

%t w[n_] := w[n] = Sum[c[n, 3 k + 2], {k, -2*n^2, 2*n^2}]

%t Table[u[n], {n, 1, z1}] (* A210698 *)

%t Table[v[n], {n, 1, z1}] (* A211071 *)

%t Table[w[n], {n, 1, z1}] (* A211071 *)

%t 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 *)

%o (Python)

%o from __future__ import division

%o def A211071(n):

%o if n % 3 == 0:

%o return 8*n**4//27

%o elif n % 3 == 1:

%o return (8*n**4 + 4*n**3 - 3*n**2 - 2*n - 7)//27

%o else:

%o return (8*n**4 + 8*n**3 - 12*n**2 - 16*n - 4)//27 # _Chai Wah Wu_, Nov 30 2016

%Y Cf. A210000, A210698, A211034.

%K nonn

%O 1,2

%A _Clark Kimberling_, Apr 01 2012