A210373 - OEIS

%I #10 Jul 16 2024 13:10:13

%S 0,3,8,48,84,243,360,768,1040,1875,2400,3888,4788,7203,8624,12288,

%T 14400,19683,22680,30000,34100,43923,49368,62208,69264,85683,94640,

%U 115248,126420,151875,165600

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

%C See A210000 for a guide to related sequences.

%H Chai Wah Wu, <a href="/A210373/b210373.txt">Table of n, a(n) for n = 0..10000</a>

%H <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 From _Chai Wah Wu_, Nov 27 2016: (Start)

%F a(n) = A210370(n)/2.

%F a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 5 -(-1)^n)*(2*n + 3 + (-1)^n)/256

%F 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 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 a = 0; b = n; z1 = 30;

%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, 2 k], {k, 0, n^2}]

%t v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]

%t w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]

%t Table[u[n], {n, 0, z1}] (* A210371 *)

%t Table[v[n], {n, 0, z1}] (* A210372 *)

%t Table[w[n], {n, 0, z1}] (* A210373 *)

%Y Cf. A210000.

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 20 2012