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%I #59 Sep 12 2016 17:00:59
%S 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,
%T 74,78,81,85,89,94,97,101,104,109,112,116,120,127,128,131,135,138,142,
%U 145,149,153,158,161,165,168,173,176,180,184,190,193,197,200,205,209
%N The infinite trunk of binary beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of 1's in binary representation of a(n).
%C 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_.
%C 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.
%H Alois P. Heinz and Antti Karttunen, <a href="/A179016/b179016.txt">Table of n, a(n) for n = 0..16405</a> (first 1000 terms from Alois P. Heinz)
%H Paul Tek, <a href="/A179016/a179016.png">Illustration of the first terms</a>
%F 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
%F a(n) = A218616(A218602(n)). - _Antti Karttunen_, Mar 04 2013
%F a(n) = A054429(A233271(A218602(n))). - _Antti Karttunen_, Dec 12 2013
%t TakeWhile[Reverse@ NestWhileList[# - DigitCount[#, 2, 1] &, 10^3, # > 0 &], # <= 209 &] (* _Michael De Vlieger_, Sep 12 2016 *)
%o (Scheme with _Antti Karttunen_'s Intseq-library for memoizing macro definec):
%o (definec (A179016 n) (cond ((< n 2) n) ((= (A218600 (A213711 n)) n) (- (expt 2 (A213711 n)) 1)) (else (- (A179016 (+ n 1)) (A213712 (+ n 1)))))) ;; _Antti Karttunen_, Nov 05 2012
%o ;; Alternatively:
%o (define (A179016 n) (A054429 (A233271 (A218602 n)))) ;; _Antti Karttunen_, Dec 12 2013
%Y Cf. A000120, A010062, A011371, A213710, A213711, A213717, A213730, A213731, A218600, A218616, A218789, A233271, A218602, A054429. First differences: A213712, complement: A213713.
%Y A subsequence of A005187, i.e., a(n) = A005187(A213715(n)). For all n,
%Y A071542(a(n)) = n, and furthermore A213708(n) <= a(n) <= A173601(n). (Cf. A218603, A218604).
%Y Rows of A218254, when reversed, converge towards this sequence.
%Y Cf. A276623, A219648, A219666, A255056, A276573, A276583, A276613 for analogous constructions, and also A259934.
%K easy,nice,nonn,base
%O 0,3
%A _Carl R. White_, Jun 24 2010
%E Starting offset changed from 1 to 0 by _Antti Karttunen_, Nov 05 2012