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A196227
Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.
5
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(0)=0 prependend by Andrew Howroyd, May 05 2020
STATUS
approved