A246064 A variation of the Farey sequence read by rows; the row of order n is defined by the greatest whole numbers less than or equal to the values generated when the numerators of the Farey sequence (order n) are replaced by n itself.

1, 1, 2, 1, 2, 3, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 6, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 7, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 7, 8, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 8

Offset: 1

Author

Neill L. Robson, Aug 24 2014

Keywords

• nonn
• tabf

Comments

Consider the area on the coordinate plane swept by a polar angle ranging from 0 to Pi/4 radians. Take every integer coordinate pair from that area whose coordinate components do not exceed the magnitude of n (i.e., if n=9, coordinate (9, 3) is chosen but not (10, 3) ). There would be a distinct number of angles between 0 and Pi/4 radians representing each point's polar angle. For large enough n, some points would share the same angle; for instance, where n=9, Cartesian points (3, 1), (6, 2), and (9, 3) would be chosen, yet all would have the radian angle arctan(1/3).

Consider these unique angle values in ascending order. Replace each angle with the number of points that could be represented in polar form with that angle (i.e., arctan(1/3) from the above example would be replaced with 3). This will result in row n of the sequence.

Row n has A005728(n) terms.

Examples

			As an example, the 4th row of the sequence (n=4) is 4, 1, 1, 2, 1, 1, 4.
Triangle starts:
  1, 1;
  2, 1, 2;
  3, 1, 1, 1, 3;
  4, 1, 1, 2, 1, 1, 4;
  5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5;
  ...
		

Crossrefs

Cf. A006843, A005728.

Programs

J

<. (% A006843)~ n
NB. A006843 is assumed to return a list of Farey denominators of order n, as described in the OEIS.

PARI

row(n) = {vf = [0]; for (k=1, n, for (m=1, k, vf = concat(vf, m/k); ); ); vf = vecsort(Set(vf)); for (i=1, #vf, print1(n\denominator(vf[i]), ", ")); } \\ Michel Marcus, Sep 04 2014

Formula

Row n of the sequence (sequence of order n) is given by floor(n/A006843(n)).