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%I #23 Nov 29 2016 08:00:36
%S 0,3,24,52,164,384,592,1131,1944,2628,4128,6144,7744,10955,15000,
%T 18100,23988,31104,36432,46179,57624,66052,81056,98304,110848,132723,
%U 157464,175284,205860,240000,264400,305723,351384,383812,438144,497664,539712,609531
%N Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 1 (mod 3).
%C Also, the number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 2 (mod 3). A211033(n) + 2*A211034(n)=n^4 for n>0. For a guide to related sequences, see A210000.
%H Chai Wah Wu, <a href="/A211034/b211034.txt">Table of n, a(n) for n = 0..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 28 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 > 12.
%F G.f.: -x*(4*x^9 + 20*x^8 + 59*x^7 + 109*x^6 + 96*x^5 + 136*x^4 + 100*x^3 + 28*x^2 + 21*x + 3)/((x - 1)^5*(x^2 + x + 1)^4).
%F If r = floor(n/3)+1, 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) = 4*n^2*(2*n^2 + 6*n + 3)/27.
%F If n == 1 mod 3, then a(n) = (8*n^4 + 28*n^3 + 33*n^2 + 16*n - 4)/27.
%F If n == 2 mod 3, then a(n) = 8*(n + 1)^4/27. (End)
%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, 0, z1}] (* A211033 *)
%t Table[v[n], {n, 0, z1}] (* A211034 *)
%t Table[w[n], {n, 0, z1}] (* A211034 *)
%t LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744}, 60] (* _Vincenzo Librandi_, Nov 29 2016 *)
%o (Python)
%o from __future__ import division
%o def A211034(n):
%o x,y,z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
%o return x**2*y**2 + 2*x**2*y*z + x**2*z**2 + 2*x*y**3 + 6*x*y**2*z + 6*x*y*z**2 + 2*x*z**3 + 2*y**3*z + 2*y*z**3 # _Chai Wah Wu_, Nov 28 2016
%Y Cf. A210000, A211033.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Mar 30 2012
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