A083368 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.

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

1,2

Comments

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

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

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

Example

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

Prog

(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

Crossrefs

A035612 is the analogous sequence for Zeckendorf representations.

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

Cf. A001511, A003714, A003754, A035612.

Cf. A000045, A000071.

Sequence in context: A002341 A348888 A128260 * A112379 A361101 A308780

Adjacent sequences: A083365 A083366 A083367 * A083369 A083370 A083371

Keyword

nonn,base,nice,easy

Author

Gary W. Adamson, Jun 04 2003

Extensions

Edited by Don Reble, Nov 12 2005

Status

approved