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A083368 a(n) is the position of the highest one in A003754(n).

%I A083368 #27 Jul 27 2025 11:15:28
%S A083368 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 A083368 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 A083368 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 A083368 a(n) is the position of the highest one in A003754(n).
%C A083368 Previous name was: 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 A083368 A003754(n), when written in binary, is the representation of n.
%C A083368 Often one uses Fibbinary representations without adjacent ones (the Zeckendorf expansion).
%C A083368 a(A000071(n+3)) = n. - _Reinhard Zumkeller_, Aug 10 2014
%C A083368 From _Gus Wiseman_, Jul 24 2025: (Start)
%C A083368 Conjecture: To obtain this sequence, start with A245563 (maximal run lengths of binary indices), then remove empty and duplicate rows (giving A385817), then take the first term of each remaining row. Some variations:
%C A083368 - For sum instead of first term we appear to have A200648.
%C A083368 - For length instead of first term we appear to have A200650+1.
%C A083368 - For last instead of first term we have A385892.
%C A083368 (End)
%D A083368 Jay Kappraff, Beyond Measure: A Guided Tour Through Nature, Myth and Number, World Scientific, 2002, page 460.
%H A083368 Reinhard Zumkeller, <a href="/A083368/b083368.txt">Table of n, a(n) for n = 1..10000</a>
%F A083368 For n = F(a)-1 to F(a+1)-2, a(n) = A035612(F(a+1)-1-n).
%F A083368 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 A083368 27 is represented 110111, 28 is 111010; the fourth position changes, so a(28)=4.
%o A083368 (Haskell)
%o A083368 a083368 n = a083368_list !! (n-1)
%o A083368 a083368_list = concat $ h $ drop 2 a000071_list where
%o A083368    h (a:fs@(a':_)) = (map (a035612 . (a' -)) [a .. a' - 1]) : h fs
%o A083368 -- _Reinhard Zumkeller_, Aug 10 2014
%Y A083368 A035612 is the analogous sequence for Zeckendorf representations.
%Y A083368 A001511 is the analogous sequence for power-of-two representations.
%Y A083368 Cf. A003714, A003754.
%Y A083368 Cf. A000045, A000071.
%Y A083368 Appears to be the first element of each row of A385817, see A083368, A200648, A200650, A385818, A385892.
%Y A083368 A000120 counts ones in binary expansion.
%Y A083368 A245563 gives run lengths of binary indices, see A089309, A090996, A245562, A246029, A328592.
%Y A083368 A384877 gives anti-run lengths of binary indices, ranks A385816.
%Y A083368 Cf. A001629, A037800, A044813, A048793, A069010, A200649, A384878, A385886.
%K A083368 nonn,base,nice,easy
%O A083368 1,2
%A A083368 _Gary W. Adamson_, Jun 04 2003
%E A083368 Edited by _Don Reble_, Nov 12 2005
%E A083368 Shorter name from _Joerg Arndt_, Jul 27 2025