A226271 -id:A226271 - cp's OEIS frontend

A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

1, 2, 3, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

M. F. Hasler, Jun 01 2013

nonn

The sequence produced by repeatedly applying t->(1+t,-1/t), starting from {1} and discarding numbers produced earlier, might be called Fibonacci or rabbit ordering of the rationals, in analogy to that ordering of the positive rationals, with t -> (1+t,1/t).

			The terms produced as described above are (grouped by iteration, including the starting value 1 = iteration 0): [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], ...

Cf. A226271, A226274, A226080, A226081, A226130.

Essentially (up to initial terms) the same as A003410, A058278, A097333 and, in particular, A226136.

o.g.f. = (1 + x + x^2 - x^3 - x^5)/(1 - x - x^3)

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

Original entry on oeis.org

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

M. F. Hasler, Jun 01 2013

nonn

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.

			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), ...

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

{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 */

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)).

cp's OEIS Frontend

A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

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