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

Author

Paul D. Hanna, Jun 21 2005

Keywords

• cofr
• nonn

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

Examples

			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.
		

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.

Programs

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

PARI

a(n)=if(n%2==1,(n+1)/2,2*a(n/2))

PARI

A109168(n)=(n+bitand(n,-n))\2 \\ M. F. Hasler, Oct 19 2019

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

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