A226081 Numerators in the Fibonacci (or rabbit) ordering of the positive rational numbers.

1, 2, 3, 1, 4, 1, 3, 5, 1, 4, 5, 2, 6, 1, 5, 7, 3, 7, 2, 5, 7, 1, 6, 9, 4, 10, 3, 7, 9, 2, 7, 8, 3, 8, 1, 7, 11, 5, 13, 4, 9, 13, 3, 10, 11, 4, 11, 2, 9, 12, 5, 11, 3, 8, 9, 1, 8, 13, 6, 16, 5, 11, 17, 4, 13, 14, 5, 16, 3, 13, 17, 7, 15, 4, 11, 13, 2, 11, 16

Offset: 1

Author

Clark Kimberling, May 25 2013

Keywords

• nonn
• frac

Comments

See A226080.

Examples

			The numerators are read from the rationals listed in "rabbit order":
1/1, 2/1, 3/1, 1/2, 4/1, 1/3, 3/2, 5/1, 1/4, 4/3, 5/2, 2/3, 6/1, ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..6000
• Index entries for fraction trees

Crossrefs

Cf. A226080.

Programs

Mathematica

z = 13; d[s_List, t_List] := Part[s, Sort[Flatten[Map[Position[s, #] &, Complement[s, t]]]]]; g[1] = {1}; g[2] = {2}; Do[ g[n] = d[Riffle[g[n - 1] + 1, 1/g[n - 1]], g[n - 2]], {n, 3, z}];  (* Edited by M. F. Hasler, Nov 30 2018 *)
j[1] = g[1]; j[n_] := Join[j[n - 1], g[n]]; j[z];  (* rabbit-ordered rationals *)
Denominator[j[z]]   (* A226080 *)
Numerator[j[z]]     (* A226081 *)

PARI

A226081_vec(N=100)={my(T=[1], S=T, A=T); while(N>#A=concat(A, apply(numerator, T=select(t->!setsearch(S, t), concat(apply(t->[t+1, 1/t], T))))), S=setunion(S, Set(T))); A} \\  M. F. Hasler, Nov 30 2018

PARI

A226081(n)=numerator(RabbitOrderedRational(n)) \\ See A226080. - M. F. Hasler, Nov 30 2018