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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

%I #6 Mar 22 2024 17:40:27

%S 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,

%T 23,6,19,11,19,16,18,7,13,14,11,26,46,7,23,12,19,22,38,27,21,39,21,33,

%U 13,27,69,57,11,19,32,36,7,26,28,11,52,92,14,23,24

%N 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.

%C 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.

%C Conjecture: every rational number in the interval (0,1] occurs exactly once in T.

%e Successive generations:

%e g(1) = (1)

%e g(2) = (1/2)

%e g(3) = (2/3, 1/4, 1/3)

%e 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)

%e Let U(n) = ordered union of g(1), g(2), ..., g(n).

%e U(1) = (1)

%e U(2) = (1, 1/2)

%e U(3) = (1, 1/2, 2/3, 1/4, 1/3)

%e 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)

%e Denominators U(4): 1, 2, 3, 4, 3, 5, 5, 4, 10, 5, 8, 9, 15, 8, 6, 20, 5, 16, 9, 15

%t (* In the remarks below, U(n) = ordered union of generations g(1), g(2),...g(n) *)

%t x = {1};

%t x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(2) *)

%t x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(3) *)

%t x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(4) *)

%t Numerator[x] (* A371279 *)

%t Denominator[x] (* this sequence *)

%t (* _Peter J. C. Moses_, Mar 16 2024 *)

%Y Cf. A226080, A371279, A371353.

%K nonn,tabf,frac

%O 1,2

%A _Clark Kimberling_, Mar 18 2024