|
| |
|
|
A371280
|
|
Irregular triangular array of denominators of the set T of fractions generated by these rules: g(1) = (1), and if x and y are in T, then x/(y+1) is in T; see Comments.
|
|
2
|
|
|
|
1, 2, 3, 4, 3, 5, 5, 4, 10, 5, 8, 9, 15, 8, 6, 20, 5, 16, 9, 15, 8, 9, 7, 13, 7, 11, 13, 23, 9, 7, 23, 6, 19, 11, 19, 16, 18, 7, 13, 14, 11, 26, 46, 7, 23, 12, 19, 22, 38, 27, 21, 39, 21, 33, 13, 27, 69, 57, 11, 19, 32, 36, 7, 26, 28, 11, 52, 92, 14, 23, 24
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Starting with g(1) = (1), write the numbers in the ordered union of g(1), g(2),…, g(n) as (x(1),x(2),…,x(m)). Then for i = 1..m, write x(i)/(1 + x(j)) for j = 1..m, and expel all the numbers that have previously occurred. The result is ordered union of g(1), g(2),..., g(n+1). The cardinalities of the first 7 unions are 1, 2, 5, 20, 245, 38179, 1032578826.
Conjecture: every rational number in the interval (0,1] occurs exactly once in T.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Successive generations:
g(1) = (1)
g(2) = (1/2)
g(3) = (2/3, 1/4, 1/3)
g(4) = (3/5, 4/5, 3/4, 3/10, 2/5, 3/8, 4/9, 8/15, 1/8, 1/6, 3/20, 1/5, 3/16, 2/9, 4/15)
Let U(n) = ordered union of g(1), g(2), ..., g(n).
U(1) = (1)
U(2) = (1, 1/2)
U(3) = (1, 1/2, 2/3, 1/4, 1/3)
U(4) = (1, 1/2, 2/3, 1/4, 1/3, 3/5, 4/5, 3/4, 3/10, 2/5, 3/8, 4/9, 8/15, 1/8, 1/6, 3/20, 1/5, 3/16, 2/9, 4/15)
Denominators U(4): 1, 2, 3, 4, 3, 5, 5, 4, 10, 5, 8, 9, 15, 8, 6, 20, 5, 16, 9, 15
|
|
|
MATHEMATICA
|
(* In the remarks below, U(n) = ordered union of generations g(1), g(2), ...g(n) *)
x = {1};
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(2) *)
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(3) *)
x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(4) *)
Denominator[x] (* this sequence *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf,frac
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
