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A210373 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive odd determinant.

%I A210373 #10 Jul 16 2024 13:10:13
%S A210373 0,3,8,48,84,243,360,768,1040,1875,2400,3888,4788,7203,8624,12288,
%T A210373 14400,19683,22680,30000,34100,43923,49368,62208,69264,85683,94640,
%U A210373 115248,126420,151875,165600
%N A210373 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive odd determinant.
%C A210373 See A210000 for a guide to related sequences.
%H A210373 Chai Wah Wu, <a href="/A210373/b210373.txt">Table of n, a(n) for n = 0..10000</a>
%H A210373 <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 A210373 From _Chai Wah Wu_, Nov 27 2016: (Start)
%F A210373 a(n) = A210370(n)/2.
%F A210373 a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 5 -(-1)^n)*(2*n + 3 + (-1)^n)/256
%F A210373 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 > 8.
%F A210373 G.f.: -x*(3*x^5 + 17*x^4 + 16*x^3 + 28*x^2 + 5*x + 3)/((x - 1)^5*(x + 1)^4). (End)
%t A210373 a = 0; b = n; z1 = 30;
%t A210373 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210373 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210373 u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
%t A210373 v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
%t A210373 w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
%t A210373 Table[u[n], {n, 0, z1}] (* A210371 *)
%t A210373 Table[v[n], {n, 0, z1}] (* A210372 *)
%t A210373 Table[w[n], {n, 0, z1}] (* A210373 *)
%Y A210373 Cf. A210000.
%K A210373 nonn
%O A210373 0,2
%A A210373 _Clark Kimberling_, Mar 20 2012