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A211034
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Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 1 (mod 3).
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4
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0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744, 10955, 15000, 18100, 23988, 31104, 36432, 46179, 57624, 66052, 81056, 98304, 110848, 132723, 157464, 175284, 205860, 240000, 264400, 305723, 351384, 383812, 438144, 497664, 539712, 609531
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,-6,6,0,4,-4,0,-1,1).
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FORMULA
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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.
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).
If r = floor(n/3)+1, 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) = 4*n^2*(2*n^2 + 6*n + 3)/27.
If n == 1 mod 3, then a(n) = (8*n^4 + 28*n^3 + 33*n^2 + 16*n - 4)/27.
If n == 2 mod 3, then a(n) = 8*(n + 1)^4/27. (End)
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MATHEMATICA
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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, 0, z1}] (* A211033 *)
Table[v[n], {n, 0, z1}] (* A211034 *)
Table[w[n], {n, 0, z1}] (* A211034 *)
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 *)
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PROG
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(Python)
from __future__ import division
x, y, z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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