|
%I #31 Sep 11 2019 23:11:56
%S 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,
%T 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,
%U 1,1,1,4,1,1,1,2,1,2,1,1,1,1,8
%N 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.
%C 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).
%C 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.
%C Row n has A005728(n) terms.
%F Row n of the sequence (sequence of order n) is given by floor(n/A006843(n)).
%e As an example, the 4th row of the sequence (n=4) is 4, 1, 1, 2, 1, 1, 4.
%e Triangle starts:
%e 1, 1;
%e 2, 1, 2;
%e 3, 1, 1, 1, 3;
%e 4, 1, 1, 2, 1, 1, 4;
%e 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5;
%e ...
%o (J) <. (% A006843)~ n
%o NB. A006843 is assumed to return a list of Farey denominators of order n, as described in the OEIS.
%o (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
%Y Cf. A006843, A005728.
%K nonn,tabf
%O 1,3
%A _Neill L. Robson_, Aug 24 2014
|
