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

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A210000, A210370.

Sequence in context: A233246 A229996 A255245 * A058920 A263472 A250287

Adjacent sequences: A210366 A210367 A210368 * A210370 A210371 A210372

Keyword

nonn

Author

Clark Kimberling, Mar 20 2012

Extensions

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

Status

approved