A210369 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even determinant.

1, 10, 65, 160, 457, 810, 1681, 2560, 4481, 6250, 9841, 12960, 18985, 24010, 33377, 40960, 54721, 65610, 84961, 100000, 126281, 146410, 181105, 207360, 252097, 285610, 342161, 384160, 454441, 506250, 592321, 655360, 759425, 835210, 959617, 1049760

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn

Comments

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

The determinant will be even if either all entries are odd or if both the leading and trailing diagonals have no more than one odd entry each. - Andrew Howroyd, Apr 28 2020

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, A210370.

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

PARI

a(n) = {((n+1)^2 - ceil(n/2)^2)^2 + ceil(n/2)^4} \\ Andrew Howroyd, Apr 28 2020

Formula

a(n) + A210370(n) = n^4.

From Colin Barker, Nov 28 2014: (Start)

a(n) = (13 + 3*(-1)^n + 4*(13+3*(-1)^n)*n + 2*(37+7*(-1)^n)*n^2 + 4*(11+(-1)^n)*n^3 + 10*n^4)/16.

G.f.: -(x^7+9*x^6+27*x^5+83*x^4+59*x^3+51*x^2+9*x+1) / ((x-1)^5*(x+1)^4).

(End)

a(n) = ((n+1)^2 - ceiling(n/2)^2)^2 + ceiling(n/2)^4. - Andrew Howroyd, Apr 28 2020

Extensions

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