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A049691 a(n)=T(n,n), array T as in A049687. Also a(n)=T(2n,2n), array T given by A049639. 24
0, 3, 5, 9, 13, 21, 25, 37, 45, 57, 65, 85, 93, 117, 129, 145, 161, 193, 205, 241, 257, 281, 301, 345, 361, 401, 425, 461, 485, 541, 557, 617, 649, 689, 721, 769, 793, 865, 901, 949, 981, 1061, 1085, 1169, 1209, 1257, 1301, 1393, 1425, 1509, 1549 (list; graph; refs; listen; history; text; internal format)
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],

of cardinalities 3,5,9,13 respectively. See also A324796/A324797. (End)

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

REFERENCES

A. O. Matveev, Farey Sequences, De Gruyter, 2017.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Visible Point

FORMULA

a(n) = A206297(n+1) = 2 + A018805(n) for n > 0. - Andrew Howroyd, Sep 17 2017

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-Fran├žois 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

CROSSREFS

Cf. A000010, A018805, A049687, A324796, A324797.

A206297 is an essentially identical sequence.

Sequence in context: A249424 A076274 A058989 * A206297 A320596 A227565

Adjacent sequences:  A049688 A049689 A049690 * A049692 A049693 A049694

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Terms a(41) and beyond from Andrew Howroyd, Sep 17 2017

STATUS

approved

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Last modified April 18 13:46 EDT 2021. Contains 343088 sequences. (Running on oeis4.)