This is Zeckendorf’s theorem, first formulated by Edouard Zeckendorf.
For our purposes here, define the Fibonacci sequence thus: , and for all . 1 and 1 are not distinct even though the first is and the latter is . We will consider two Fibonacci numbers and consecutive if their indexes and are consecutive integers, e.g., .
where is the th element in . This ordered tuplet is the Zeckendorf representation of , or we might even say the Fibonacci base representation of (or the Fibonacci coding of ).
So for example, 53 = 34 + 13 + 5 + 1, that is, . Furthermore, . We list the constituent elements in descending order from to to facilitate reinterpretation as a binary integer, 10101001 (or 169) in this example. Taking the Zeckendorf representations of integers in order and reinterpreting in binary as
gives the sequence 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, … (A003714 in Sloane’s OEIS). It can be observed that these numbers have no consecutive 1s in their binary representations.
|Date of creation||2013-03-22 16:03:57|
|Last modified on||2013-03-22 16:03:57|
|Last modified by||CompositeFan (12809)|