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A211068
Number of 2 X 2 matrices having all terms in {1,...,n} and positive odd determinant.
4
0, 3, 20, 48, 144, 243, 528, 768, 1400, 1875, 3060, 3888, 5880, 7203, 10304, 12288, 16848, 19683, 26100, 30000, 38720, 43923, 55440, 62208, 77064, 85683, 104468, 115248, 138600, 151875, 180480, 196608, 231200, 250563, 291924
OFFSET
1,2
COMMENTS
For a guide to related sequences, see A210000.
FORMULA
From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = A211065(n)/2.
a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 1 -(-1)^n)*(2*n - 1 + (-1)^n)/256.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: -x^2*(3*x^5 + 5*x^4 + 28*x^3 + 16*x^2 + 17*x + 3)/((x - 1)^5*(x + 1)^4). (End)
MATHEMATICA
a = 1; b = n; z1 = 35;
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, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}] (* A211066 *)
Table[v[n], {n, 1, z1}] (* A211067 *)
Table[w[n], {n, 1, z1}] (* A211068 *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 3, 20, 48, 144, 243, 528, 768, 1400}, 50] (* Vincenzo Librandi, Nov 28 2016 *)
PROG
(Magma) [(2*n+1-(-1)^n)^2*(6*n+1-(-1)^n)*(2*n-1+(-1)^n)/256: n in [1..40]]; // Vincenzo Librandi, Nov 28 2016
CROSSREFS
Cf. A210000.
Sequence in context: A072472 A267055 A296252 * A343010 A360417 A281268
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 31 2012
STATUS
approved