A226274 Position of 1/n in the ordering of the rationals given by application of the map t -> (1+t,-1/t), cf. A226130.

1, 9, 31, 100, 317, 1000, 3150, 9918, 31223, 98289, 309406, 973981, 3065996, 9651448, 30381786, 95638797, 301061279, 947710512, 2983297009, 9391117780, 29562290606, 93059106094, 292940670339, 922147653681, 2902827709802, 9137808548505, 28764898718296, 90548996937472

Offset: 1

Author

M. F. Hasler, Jun 01 2013

Keywords

• nonn

Comments

An analog of the Fibonacci ordering of the positive rationals (cf. A226080) is the sequence of rationals produced from the initial vector [1] by appending iteratively the new rationals obtained by applying the map t-> (t+1, -1/t) to each term of the vector (cf. example).

It is seen that the unit fraction 1/n appears as the last term produced in the (3n-3)th iteration, therefore the indices a(n) equal every third terms in the partial sums of A226275 (= new terms produced during the respective iteration), cf. formula.

Examples

			Starting with [1], applying the map t->(1+t,-1/t) to the (most recently obtained) vector and discarding the numbers occurring earlier, one gets the sequence (grouped by "generation"): [1], [2, -1], [3, -1/2, 0], [4, -1/3, 1/2], [5, -1/4, 2/3, 3/2, -2], [6, -1/5, 3/4, 5/3, -3/2, 5/2, -2/3], [7, -1/6, 4/5, 7/4, -4/3, 8/3, -3/5, 7/2, -2/5, 1/3], [8, -1/7, 5/6, 9/5, -5/4, 11/4, -4/7, 11/3, -3/8, 2/5, 9/2, -2/7, 3/5, 4/3, -3],...
The unit fractions 1/1, 1/2, 1/3, 1/4,... occur at positions 1, 9(=1+2+3+3), 31(=9+5+7+10), 100(=31+15+22+32), ...
		

Crossrefs

Cf. A226271, A226275, A226080, A226081, A226130.

Programs

PARI

{print1([s=1]", ");U=Set(g=[1]); for(n=1,29,U=setunion(U,Set(g=select(f->!setsearch(U,f), concat(apply(t->[t+1,if(t,-1/t)],g))))); for(i=1,#g, numerator(g[i])==1&&print1(s+i/*",g[i],*/","));s+=#g)} /* illustrative purpose only */

Formula

a(n) = s(3n-3) where s(k) = Sum_{j=0..k} A226275(j).

O.g.f.: x(1 + 4*x - 7*x^2 + 4*x^3 - x^4)/((1 - x)(1 - 4*x + 3*x^2 - x^3)).