A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.

0, 6, 16, 96, 168, 486, 720, 1536, 2080, 3750, 4800, 7776, 9576, 14406, 17248, 24576, 28800, 39366, 45360, 60000, 68200, 87846, 98736, 124416, 138528, 171366, 189280, 230496, 252840, 303750, 331200, 393216, 426496, 501126, 541008, 629856, 677160, 781926, 837520

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn

Comments

a(n) is also the number of 2 X 2 matrices with all elements in {0,1,...n} and odd permanent.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).

Crossrefs

Cf. A210000, A210369.

Programs

Mathematica

a = 0; b = n; z1 = 28;
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_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 0,  z1}] (* A210369 *)
Table[v[n], {n, 0, z1}](* A210370 *)

PARI

a(n)={2*((n+1)^2-ceil(n/2)^2)*ceil(n/2)^2} \\ Andrew Howroyd, Apr 28 2020

Formula

a(n) + A210369(n) = n^4.

From Colin Barker, Nov 28 2014: (Start)

a(n) = (3 - 3*(-1)^n - 12*(-1+(-1)^n)*n + (22-14*(-1)^n)*n^2 - 4*(-5+(-1)^n)*n^3 + 6*n^4)/16.

G.f.: -2*x*(3*x^5+17*x^4+16*x^3+28*x^2+5*x+3) / ((x-1)^5*(x+1)^4).

(End)

a(n) = 2*((n+1)^2 - ceiling(n/2)^2)*ceiling(n/2)^2. - Andrew Howroyd, Apr 28 2020

Extensions

Terms a(29) and beyond from Andrew Howroyd, Apr 28 2020