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

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A210000, A210369.

Sequence in context: A009352 A056204 A091148 * A266180 A229566 A222965

Adjacent sequences: A210367 A210368 A210369 * A210371 A210372 A210373

Keyword

nonn

Author

Clark Kimberling, Mar 20 2012

Extensions

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

Status

approved