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A Fibbinary system represents a number as a sum of distinct Fibonacci numbers (instead of distinct powers of two). Using representations without adjacent zeros, a(n) = the highest bit-position which changes going from n-1 to n.
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%I #18 Oct 19 2017 03:14:16

%S 1,2,1,3,2,1,4,1,3,2,1,5,2,1,4,1,3,2,1,6,1,3,2,1,5,2,1,4,1,3,2,1,7,2,

%T 1,4,1,3,2,1,6,1,3,2,1,5,2,1,4,1,3,2,1,8,1,3,2,1,5,2,1,4,1,3,2,1,7,2,

%U 1,4,1,3,2,1,6,1,3,2,1,5,2,1,4,1,3,2,1,9,2,1,4,1,3,2,1,6,1,3,2,1,5,2

%N A Fibbinary system represents a number as a sum of distinct Fibonacci numbers (instead of distinct powers of two). Using representations without adjacent zeros, a(n) = the highest bit-position which changes going from n-1 to n.

%C A003754(n), when written in binary, is the representation of n.

%C Often one uses Fibbinary representations without adjacent ones (the Zeckendorf expansion).

%C a(A000071(n+3)) = n. - _Reinhard Zumkeller_, Aug 10 2014

%D Jay Kappraff, Beyond Measure: A Guided Tour Through Nature, Myth and Number, World Scientific, 2002, page 460.

%H Reinhard Zumkeller, <a href="/A083368/b083368.txt">Table of n, a(n) for n = 1..10000</a>

%F For n = F(a)-1 to F(a+1)-2, a(n) = A035612(F(a+1)-1-n).

%F a(n) = a(k)+1 if n = ceiling(phi*k) where phi is the golden ratio; otherwise a(n) = 1. - _Tom Edgar_, Aug 25 2015

%e 27 is represented 110111, 28 is 111010; the fourth position changes, so a(28)=4.

%o (Haskell)

%o a083368 n = a083368_list !! (n-1)

%o a083368_list = concat $ h $ drop 2 a000071_list where

%o h (a:fs@(a':_)) = (map (a035612 . (a' -)) [a .. a' - 1]) : h fs

%o -- _Reinhard Zumkeller_, Aug 10 2014

%Y A035612 is the analogous sequence for Zeckendorf representations.

%Y A001511 is the analogous sequence for power-of-two representations.

%Y Cf. A001511, A003714, A003754, A035612.

%Y Cf. A000045, A000071.

%K nonn,base,nice,easy

%O 1,2

%A _Gary W. Adamson_, Jun 04 2003

%E Edited by _Don Reble_, Nov 12 2005