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

%I A196227 #21 May 05 2020 17:07:33
%S A196227 0,0,2,8,14,28,34,56,70,92,106,144,158,204,226,256,286,348,370,440,
%T A196227 470,516,554,640,670,748,794,864,910,1020,1050,1168,1230,1308,1370,
%U A196227 1464,1510,1652,1722,1816,1878,2036,2082,2248,2326,2420,2506,2688,2750,2916,2994
%N A196227 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.
%C A196227 It is also the number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is -1.
%H A196227 Andrew Howroyd, <a href="/A196227/b196227.txt">Table of n, a(n) for n = 0..1000</a>
%F A196227 From _Andrew Howroyd_, May 05 2020: (Start)
%F A196227 a(n) = A171503(n) - (2*n + 1) for n > 0.
%F A196227 a(n) = -2*(n + 1) + 4*Sum_{k=1..n} phi(k) for n > 0.
%F A196227 a(n) = 2 * A209978(n). (End)
%p A196227 a:= proc(n) option remember; `if`(n<2, 0,
%p A196227       a(n-1)-2 + 4*numtheory[phi](n))
%p A196227     end:
%p A196227 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 05 2020
%t A196227 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 *)
%o A196227 (PARI) a(n) = if(n < 1, 0, 4*sum(k=1, n, eulerphi(k)) - 2*(n + 1)) \\ _Andrew Howroyd_, May 05 2020
%Y A196227 Cf. A171503 (determinants of matrices that include zero), A209978, A210000.
%K A196227 nonn
%O A196227 0,3
%A A196227 _Aldo González Lorenzo_, Sep 29 2011
%E A196227 a(0)=0 prependend by _Andrew Howroyd_, May 05 2020