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A211068 Number of 2 X 2 matrices having all terms in {1,...,n} and positive odd determinant.

%I A211068 #15 Sep 08 2022 08:46:01
%S A211068 0,3,20,48,144,243,528,768,1400,1875,3060,3888,5880,7203,10304,12288,
%T A211068 16848,19683,26100,30000,38720,43923,55440,62208,77064,85683,104468,
%U A211068 115248,138600,151875,180480,196608,231200,250563,291924
%N A211068 Number of 2 X 2 matrices having all terms in {1,...,n} and positive odd determinant.
%C A211068 For a guide to related sequences, see A210000.
%H A211068 Chai Wah Wu, <a href="/A211068/b211068.txt">Table of n, a(n) for n = 1..10000</a>
%H A211068 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,-6,6,4,-4,-1,1).
%F A211068 From _Chai Wah Wu_, Nov 27 2016: (Start)
%F A211068 a(n) = A211065(n)/2.
%F A211068 a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 1 -(-1)^n)*(2*n - 1 + (-1)^n)/256.
%F A211068 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.
%F A211068 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)
%t A211068 a = 1; b = n; z1 = 35;
%t A211068 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A211068 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A211068 u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
%t A211068 v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
%t A211068 w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
%t A211068 Table[u[n], {n, 1, z1}]  (* A211066 *)
%t A211068 Table[v[n], {n, 1, z1}]  (* A211067 *)
%t A211068 Table[w[n], {n, 1, z1}]  (* A211068 *)
%t 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 *)
%o A211068 (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
%Y A211068 Cf. A210000.
%K A211068 nonn
%O A211068 1,2
%A A211068 _Clark Kimberling_, Mar 31 2012