A072193 Concatenate continued fraction expansions of the rational numbers 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ...

2, 3, 1, 2, 4, 2, 1, 3, 5, 2, 2, 1, 1, 2, 1, 4, 6, 3, 2, 1, 2, 1, 5, 7, 3, 2, 2, 3, 1, 1, 3, 1, 2, 2, 1, 6, 8, 4, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 7, 9, 4, 2, 3, 2, 4, 1, 1, 4, 1, 2, 1, 3, 2, 1, 8, 10, 5, 3, 3, 2, 2, 2, 1, 1, 2, 1, 2, 3, 1, 4, 1, 9, 11, 5, 2, 3, 1, 2, 2, 1, 3, 2, 5, 1, 1, 5, 1, 1, 1, 3, 1, 2

Offset

0,1

Comments

Leading zeros in continued fraction omitted.

The geometric mean of the sequence equals Khintchine's constant K=2.685452001 = A002210 since the frequency of the integers agrees with the Gauss-Kuzmin distribution. - Jwalin Bhatt, Nov 24 2025

References

K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Math. Assoc. America, 2002, p. 72.

Links

Robert Israel, Table of n, a(n) for n = 0..10299

Example

Table starts
  2
  3
  1  2
  4
  2
  1  3
  5
  2  2
  1  1  2. - Robert Israel, Sep 18 2015

Maple

seq(seq(op(cfrac(i/j, 'quotients')[2..-1]), i=1..j-1), j=2..20); # Robert Israel, Sep 18 2015

Mathematica

Table[Rest@ ContinuedFraction[k/n], {n, 2, 11}, {k, n - 1}] // Flatten (* Michael De Vlieger, Sep 18 2015 *)

Prog

(PARI) {m=11; for(i=2, m, for(j=1, i-1, c=contfrac(j/i); for(k=2, matsize(c)[2], print1(c[k], ", "))))}
(Python)
from sympy import Rational, continued_fraction_iterator
A072193 = [cf for i in range(2, 12) for j in range(1, i) for cf in continued_fraction_iterator(Rational(i, j))]  # Jwalin Bhatt, Nov 24 2025

Crossrefs

Cf. A002210, A084580, A390946.

Sequence in context: A344089 A329631 A239304 * A233359 A279345 A097966

Adjacent sequences: A072190 A072191 A072192 * A072194 A072195 A072196

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane, Nov 10 2002

Extensions

Extended by Klaus Brockhaus and Vladeta Jovovic, Nov 13 2002

Status

approved