OFFSET
0,2
COMMENTS
a(n) is related to the sequence b(n) = |{(x, y): gcd(x, y) = 1, 1<=x, y<=n}| (A018805) as follows: a(n) = b(n - 1) + 2 (for n > 1). - Shawn Westmoreland (westmore(AT)math.utexas.edu), Jun 11 2003
Comment from N. J. A. Sloane, Sep 08 2019 (Start)
The above comment can be rephrased as saying that a(n) is the cardinality of the subsequence F(B(2n),n) of the Farey series F_{2n} that is extensively studied in Matveev (2017). See the definition on page 1.
For example, F(B(2),1), F(B(4),2), F(B(6),3), and F(B(8),4) are:
[0, 1/2, 1],
[0, 1/3, 1/2, 2/3, 1],
[0, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1],
[0, 1/5, 1/4, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 3/4, 4/5, 1],
a(n) is the number of visible points on an n X n square lattice when viewed from (0, 0), (0, n), (n, 0), or (n, n). - Torlach Rush, Nov 16 2020
Also number of elements in { c/d ; -d <= c <= d <= n }, i.e., distinct fractions with denominator not exceeding n and absolute value of numerator not exceeding the denominator. - M. F. Hasler, Mar 26 2023
REFERENCES
A. O. Matveev, Farey Sequences, De Gruyter, 2017.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
A. O. Matveev, Farey Sequences: Errata + Haskell code
Eric Weisstein's World of Mathematics, Visible Point
FORMULA
a(n) = 1 + 2 * Sum{k=1..n} A000010(k), n > 0. - Torlach Rush, Nov 24 2020
MAPLE
Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end: # A006842/A006843
BF := proc(m) local a, i, h, k; global Farey; a:=[];
for i in Farey(2*m) do h:=numer(i); k:=denom(i);
if (h <= m) and (k-m <= h) then a:=[op(a), i]; fi; od: a; end;
[seq(nops(BF(m), m=1..20)]; # this sequence - N. J. A. Sloane, Sep 08 2019
MATHEMATICA
a[0] = 0; a[n_] := 2 + Sum[Quotient[n, g]^2*MoebiusMu[g], {g, 1, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 07 2017, translated from PARI *)
PROG
(PARI) a(n) = if(n>0, 2, 0) + sum(g=1, n, (n\g)^2 * moebius(g)); \\ Andrew Howroyd, Sep 17 2017
(PARI) a(n) = if(n>0, 1, 0) + 2 * sum(k=1, n, eulerphi(k)); \\ Torlach Rush, Nov 24 2020
(PARI) a(n)=#Set(concat([[c/d|c<-[-d..d], d]|d<-[0..n]])) \\ For illustrative purpose only! - M. F. Hasler, Mar 26 2023
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A049691(n):
if n == 0:
return 0
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*(A049691(k1)-2)
j, k1 = j2, n//j2
return n*(n-1)-c+j+2 # Chai Wah Wu, Aug 04 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Terms a(41) and beyond from Andrew Howroyd, Sep 17 2017
STATUS
approved