A209993 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant in {-1,0,1}.

1, 16, 45, 94, 159, 248, 349, 478, 623, 792, 973, 1182, 1423, 1672, 1933, 2238, 2559, 2888, 3261, 3630, 4063, 4504, 4925, 5374, 5935, 6456, 6957, 7534, 8159, 8728, 9453, 10062, 10767, 11480, 12141, 12942, 13855, 14584, 15325, 16174, 17183

Offset: 0

Author

Clark Kimberling, Mar 18 2012

Keywords

• nonn

Comments

See A210000 for a guide to related sequences.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000010, A018804, A210000.

Programs

Maple

pillai:= proc(n) local i; add(igcd(i,n),i=1..n) end proc:
T:= 16: R:= 1,16:
for n from 2 to 50 do
  v:= 1 +  4*n + 8*numtheory:-phi(n) + 4*pillai(n);
  T:= T + v;
  R:= R,T;
od:
R; # Robert Israel, Jan 07 2024

Mathematica

a = 1; b = n; z1 = 40;
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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, 1}]
Table[c1[n, 1], {n, 0, z1}]   (* A209992 *)

Formula

For n > 1, a(n) - a(n-1) = 1 + 4 * n + 8 * A000010(n) + 4 * A018804(n). - Robert Israel, Jan 07 2024