A196227 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.

0, 0, 2, 8, 14, 28, 34, 56, 70, 92, 106, 144, 158, 204, 226, 256, 286, 348, 370, 440, 470, 516, 554, 640, 670, 748, 794, 864, 910, 1020, 1050, 1168, 1230, 1308, 1370, 1464, 1510, 1652, 1722, 1816, 1878, 2036, 2082, 2248, 2326, 2420, 2506, 2688, 2750, 2916, 2994

Offset: 0

Author

Aldo González Lorenzo, Sep 29 2011

Keywords

• nonn

Comments

It is also the number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is -1.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A171503 (determinants of matrices that include zero), A209978, A210000.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)-2 + 4*numtheory[phi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 05 2020

Mathematica

Table[cnt = 0; Do[If[a*d-b*c == 1, cnt++], {a, n}, {b, n}, {c, n}, {d, n}]; cnt, {n, 50}] (* T. D. Noe, Oct 11 2011 *)

PARI

a(n) = if(n < 1, 0, 4*sum(k=1, n, eulerphi(k)) - 2*(n + 1)) \\ Andrew Howroyd, May 05 2020

Formula

From Andrew Howroyd, May 05 2020: (Start)

a(n) = A171503(n) - (2*n + 1) for n > 0.

a(n) = -2*(n + 1) + 4*Sum_{k=1..n} phi(k) for n > 0.

a(n) = 2 * A209978(n). (End)

Extensions

a(0)=0 prependend by Andrew Howroyd, May 05 2020