A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

1, 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, 9, 10, 10, 12, 11, 12, 12, 16, 13, 14, 14, 16, 15, 16, 16, 32, 17, 18, 18, 20, 19, 20, 20, 24, 21, 22, 22, 24, 23, 24, 24, 32, 25, 26, 26, 28, 27, 28, 28, 32, 29, 30, 30, 32, 31, 32, 32, 64, 33, 34, 34, 36, 35, 36, 36, 40, 37, 38, 38

Offset

1,2

Comments

Compare with continued fraction A100338.

The sequence is equal to the sequence of positive integers (1, 2, 3, 4, ...) interleaved with the sequence multiplied by two, 2*(1, 2, 2, 4, 3, ...) = (2, 4, 4, 8, 6, ...): see the first formula. - M. F. Hasler, Oct 19 2019

Links

Table of n, a(n) for n=1..75.

Formula

a(2*n-1) = n, a(2*n) = 2*a(n) for all n >= 1.

a((2*n-1)*2^p) = n * 2^p, p >= 0. - Johannes W. Meijer, Jun 22 2011

a(n) = n - (n AND n-1)/2. - Gary Detlefs, Jul 10 2014

a(n) = A285326(n)/2. - Antti Karttunen, Apr 19 2017

a(n) = A140472(n). - M. F. Hasler, Oct 19 2019

Example

x=1.408494279228906985748474279080697991613998955782051281466263817524862977...
The continued fraction expansion of 2*x = A109170:
[2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]
which equals the continued fraction of x interleaved with the even numbers.

Maple

nmax:=75; pmax:= ceil(log(nmax)/log(2)); for p from 0 to pmax do for n from 1 to nmax do a((2*n-1)*2^p):= n*2^p: od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011

Prog

(PARI) a(n)=if(n%2==1, (n+1)/2, 2*a(n/2))
(Scheme, with memoization-macro definec)
(definec (A109168 n) (if (zero? n) n (if (odd? n) (/ (+ 1 n) 2) (* 2 (A109168 (/ n 2))))))
;; Antti Karttunen, Apr 19 2017
(PARI) A109168(n)=(n+bitand(n, -n))\2 \\ M. F. Hasler, Oct 19 2019

Crossrefs

Cf. A109169 (digits of x), A109170 (continued fraction of 2*x), A109171 (digits of 2*x).

Cf. A006519 and A129760. [Johannes W. Meijer, Jun 22 2011]

Half the terms of A285326.

Sequence in context: A086835 A046701 A140472 * A015134 A171580 A246796

Adjacent sequences: A109165 A109166 A109167 * A109169 A109170 A109171

Keyword

cofr,nonn

Author

Paul D. Hanna, Jun 21 2005

Status

approved