OFFSET
0,3
COMMENTS
The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):
- in the Zeckendorf representation (or greedy Fibonacci representation):
- Fibonacci numbers are as big as possible (see A035517),
- and the corresponding binary encoding, A003714(n),
cannot have two consecutive 1's;
- in the dual Zeckendorf representation (or lazy Fibonacci representation):
- Fibonacci numbers are as small as possible (see A112309),
- and the corresponding binary encoding, A003754(n+1),
cannot have two consecutive nonleading 0's.
See A356326 for a similar sequence.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program
FORMULA
EXAMPLE
For n = 28:
- using F(k) = A000045(k),
- the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
- the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
- F(5) and F(3) appear in both representations,
- so a(28) = F(5) + F(3) = 7.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Aug 27 2022
STATUS
approved