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%I #15 Aug 10 2015 07:26:29
%S 0,1,2,4,5,7,9,12,14,17,20,22,24,27,29,33,35,37,40,42,45,47,50,54,56,
%T 58,61,63,67,70,74,76,79,83,88,90,92,95,97,101,104,108,110,113,117,
%U 121,123,126,130,134,138,143,145,147,150,152,156,159,163,165,168
%N The infinite trunk of Zeckendorf beanstalk. The only infinite sequence such that a(n-1) = a(n) - number of 1's in Zeckendorf 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 "Zeckendorf beanstalk" from its root (zero).
%C There are many finite sequences such as 0,1,2; 0,1,2,4,5; etc. (see A219649) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
%C There can be only one infinite trunk in "Zeckendorf beanstalk" as all paths downwards from numbers >= A000045(i) must pass through A000045(i)-1 (i.e. A000071(i)). This provides also a well-defined method to compute the sequence, for example, via a partially reversed version A261076.
%C See A014417 for the Fibonacci number system representation, also known as Zeckendorf expansion.
%H Antti Karttunen, <a href="/A219648/b219648.txt">Table of n, a(n) for n = 0..11817</a>
%F a(n) = A261076(A261102(n)).
%o (Scheme) (define (A219648 n) (A261076 (A261102 n)))
%Y Cf. A000045, A000071, A007895, A014417, A219641, A219649, A261076, A261102. For all n, A219642(a(n)) = n and A219643(n) <= a(n) <= A219645(n). Cf. also A261083 & A261084.
%Y Other similarly constructed sequences: A179016, A219666, A255056.
%K nonn
%O 0,3
%A _Antti Karttunen_, Nov 24 2012
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