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

0,3

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

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A056677 A053697 A092346 * A166045 A294632 A073665

Adjacent sequences: A196224 A196225 A196226 * A196228 A196229 A196230

Keyword

nonn

Author

Aldo González Lorenzo, Sep 29 2011

Extensions

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

Status

approved