

A179016


The infinite trunk of binary beanstalk: The only infinite sequence such that a(n1) = a(n)  number of 1's in binary representation of a(n).


89



0, 1, 3, 4, 7, 8, 11, 15, 16, 19, 23, 26, 31, 32, 35, 39, 42, 46, 49, 53, 57, 63, 64, 67, 71, 74, 78, 81, 85, 89, 94, 97, 101, 104, 109, 112, 116, 120, 127, 128, 131, 135, 138, 142, 145, 149, 153, 158, 161, 165, 168, 173, 176, 180, 184, 190, 193, 197, 200, 205, 209
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OFFSET

0,3


COMMENTS

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "binary beanstalk" from its root (zero). The name "beanstalk" is due to Antti Karttunen.
There are many finite sequences such as 0,1,2; 0,1,3,4,7,9; etc. obeying the same condition (see A218254) and as the length increases, so (necessarily) does the similarity to this infinite sequence.


LINKS



FORMULA

a(0)=0, a(1)=1, and for n > 1, if n = A218600(A213711(n)) then a(n) = (2^A213711(n))  1, and in other cases, a(n) = a(n+1)  A213712(n+1). (This formula is based on Carl White's observation that this iterated/converging path must pass through each (2^n)1. However, it would be very interesting to know whether the sequence admits more traditional recurrence(s), referring to previous, not to further terms in the sequence in their definition!)  Antti Karttunen, Oct 26 2012


MATHEMATICA

TakeWhile[Reverse@ NestWhileList[#  DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* Michael De Vlieger, Sep 12 2016 *)


PROG

(Scheme with Antti Karttunen's Intseqlibrary for memoizing macro definec):
;; Alternatively:


CROSSREFS

Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
Rows of A218254, when reversed, converge towards this sequence.


KEYWORD

easy,nice,nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



