# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a371279 Showing 1-1 of 1 %I A371279 #5 Mar 22 2024 17:40:18 %S A371279 1,1,2,1,1,3,4,3,3,2,3,4,8,1,1,3,1,3,2,4,5,5,4,10,5,8,9,15,8,6,20,5, %T A371279 16,9,15,5,5,2,5,5,4,9,15,3,10,5,8,9,15,10,8,20,10,16,6,16,40,32,6,10, %U A371279 5,5,1,5,5,2,9,15,3,5,5,4,9,15,5,4,10,5,8,3 %N A371279 Irregular triangular array of numerators 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 A371279 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 A371279 Conjecture: every rational number in the interval (0,1] occurs exactly once in T. %e A371279 Successive generations: %e A371279 g(1) = (1) %e A371279 g(2) = (1/2) %e A371279 g(3) = (2/3, 1/4, 1/3) %e A371279 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 A371279 Let U(n) = ordered union of g(1), g(2), ..., g(n). %e A371279 U(1) = (1) %e A371279 U(2) = (1, 1/2) %e A371279 U(3) = (1, 1/2, 2/3, 1/4, 1/3) %e A371279 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 A371279 Numerators in U(4): 1, 1, 2, 1, 1, 3, 4, 3, 3, 2, 3, 4, 8, 1, 1, 3, 1, 3, 2, 4. %t A371279 (* In the remarks below, U(n) = ordered union of generations g(1), g(2),...g(n) *) %t A371279 x = {1}; %t A371279 x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(2) *) %t A371279 x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(3) *) %t A371279 x = DeleteDuplicates[Join[x, Map[x[[#[[1]]]]/(1 + x[[#[[2]]]]) &, Tuples[Range[Length[x]], {2}]]]] (* U(4) *) %t A371279 Numerator[x] (* this sequence *) %t A371279 Denominator[x] (* A371280 *) %t A371279 (* _Peter J. C. Moses_, Mar 16 2024 *) %Y A371279 Cf. A226080, A371280. %K A371279 nonn,tabf,frac %O A371279 1,3 %A A371279 _Clark Kimberling_, Mar 18 2024 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE