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

0,2

Comments

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).

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Formula

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

Example

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], ...

Mathematica

CoefficientList[Series[(1+x+x^2-x^3-x^5)/(1-x-x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{1, 0, 1}, {1, 2, 3, 3, 5, 7}, 50] (* Harvey P. Dale, Mar 04 2026 *)

Prog

(PARI) a(n)=if(n>2, ([0, 1, 0; 0, 0, 1; 1, 0, 1]^(n-3)*[3; 5; 7])[1, 1], n+1) \\ Charles R Greathouse IV, May 28 2026

Crossrefs

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

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

Sequence in context: A017984 A286330 A241378 * A035066 A258967 A035068

Adjacent sequences: A226272 A226273 A226274 * A226276 A226277 A226278

Keyword

nonn,easy

Author

M. F. Hasler, Jun 01 2013

Status

approved