

A109168


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


7



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Compare with continued fraction A100338.


LINKS

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


FORMULA

For n>=1: a(2*n1) = n, a(2*n) = 2*a(n).
a((2*n1)*2^p) = n * 2^p, p>=0. [Johannes W. Meijer, Jun 22 2011]
a(n) = n  (n AND n1)/2.  Gary Detlefs, Jul 10 2014
a(n) = A285326(n)/2.  Antti Karttunen, Apr 19 2017


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 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*n1)*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 memoizationmacro definec)
(definec (A109168 n) (if (zero? n) n (if (odd? n) (/ (+ 1 n) 2) (* 2 (A109168 (/ n 2))))))
;; Antti Karttunen, Apr 19 2017


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



