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

0, 3, 8, 48, 84, 243, 360, 768, 1040, 1875, 2400, 3888, 4788, 7203, 8624, 12288, 14400, 19683, 22680, 30000, 34100, 43923, 49368, 62208, 69264, 85683, 94640, 115248, 126420, 151875, 165600

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn

Comments

See A210000 for a guide to related sequences.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).

Crossrefs

Cf. A210000.

Programs

Mathematica

a = 0; b = n; z1 = 30;
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

Formula

From Chai Wah Wu, Nov 27 2016: (Start)

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

a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 5 -(-1)^n)*(2*n + 3 + (-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 > 8.

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)