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

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Jun 04 2003

Keywords

• nonn
• base
• nice
• easy

Comments

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.

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

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

a(A000071(n+3)) = n. - Reinhard Zumkeller, Aug 10 2014

From Gus Wiseman, Jul 24 2025: (Start)

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:

- For sum instead of first term we appear to have A200648.

- For length instead of first term we appear to have A200650+1.

- For last instead of first term we have A385892.

(End)

Examples

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

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

A035612 is the analogous sequence for Zeckendorf representations.

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

Cf. A003714, A003754.

Cf. A000045, A000071.

Appears to be the first element of each row of A385817, see A083368, A200648, A200650, A385818, A385892.

A000120 counts ones in binary expansion.

A245563 gives run lengths of binary indices, see A089309, A090996, A245562, A246029, A328592.

A384877 gives anti-run lengths of binary indices, ranks A385816.

Cf. A001629, A037800, A044813, A048793, A069010, A200649, A384878, A385886.

Programs

Haskell

a083368 n = a083368_list !! (n-1)
a083368_list = concat $ h $ drop 2 a000071_list where
   h (a:fs@(a':_)) = (map (a035612 . (a' -)) [a .. a' - 1]) : h fs
-- Reinhard Zumkeller, Aug 10 2014

Formula

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

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

Extensions

Edited by Don Reble, Nov 12 2005

Shorter name from Joerg Arndt, Jul 27 2025