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

%I A209993 #17 Jan 08 2024 09:03:59
%S A209993 1,16,45,94,159,248,349,478,623,792,973,1182,1423,1672,1933,2238,2559,
%T A209993 2888,3261,3630,4063,4504,4925,5374,5935,6456,6957,7534,8159,8728,
%U A209993 9453,10062,10767,11480,12141,12942,13855,14584,15325,16174,17183
%N A209993 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant in {-1,0,1}.
%C A209993 See A210000 for a guide to related sequences.
%H A209993 Robert Israel, <a href="/A209993/b209993.txt">Table of n, a(n) for n = 0..10000</a>
%F A209993 For n > 1, a(n) - a(n-1) = 1 + 4 * n + 8 * A000010(n) + 4 * A018804(n). - _Robert Israel_, Jan 07 2024
%p A209993 pillai:= proc(n) local i; add(igcd(i,n),i=1..n) end proc:
%p A209993 T:= 16: R:= 1,16:
%p A209993 for n from 2 to 50 do
%p A209993   v:= 1 +  4*n + 8*numtheory:-phi(n) + 4*pillai(n);
%p A209993   T:= T + v;
%p A209993   R:= R,T;
%p A209993 od:
%p A209993 R; # _Robert Israel_, Jan 07 2024
%t A209993 a = 1; b = n; z1 = 40;
%t A209993 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A209993 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A209993 c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, 1}]
%t A209993 Table[c1[n, 1], {n, 0, z1}]   (* A209992 *)
%Y A209993 Cf. A000010, A018804, A210000.
%K A209993 nonn
%O A209993 0,2
%A A209993 _Clark Kimberling_, Mar 18 2012