OFFSET
1,5
COMMENTS
The parameter k is the square of the Euclidean distance of the corresponding points to the origin in the "denominator, numerator" representation space.
The fractions in order begin: 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 1/6, 4/5, 1/7, 2/7, 3/7, 5/6, ..., .
Note that the fraction 2/4 is not in the above since it can be reduced to 1/2.
EXAMPLE
The point (1,0) lies closer to the origin than any other point in the first quadrant. Therefore a(1) = 0;
The next point is (1,1) because it lies closer to the origin. Therefore a(2) = 1;
The point (2,1) lies the next closest to the origin. Therefore a(3) = 1;
The point (3,1) is the next closest to the origin. Therefore a(4)= 1;
The point (3,2) is the next closest to the origin. Therefore a(5)= 2;
etc.
The fractions 4/7 and 1/8 are respectively represented by the points (7,4) and (8,1), both points are at the same distance sqrt(65) from the origin, but 4/7 has smaller denominator than 1/8 hence a(17)=4 and a(18)=1.
MATHEMATICA
nmax = 22; (* Maximum explorative denominator *)
fracs = Sort@Union@Flatten@Table[a/b, {b, nmax}, {a, 0, b}];
(* Sorting generated fractions according to increasing Euclidean distance first, and then by increasing denominator *)
fracsorted = SortBy[fracs, {(Numerator@#)^2 + (Denominator@#)^2 &,
Denominator@# &}];
nmaxlimit = Floor[nmax^2/4] (*Safe limit for a correctly sorted sequence*);
Take[Numerator@fracsorted, nmaxlimit]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Robert G. Wilson v and Andres Cicuttin, Dec 18 2016
STATUS
approved