Concrete mathematics : a foundation for computer science

282 SPECIAL NUMBERS
Fk < n < Fk+l; then we have 0 6 n - Fk < Fk+l -- Fk = Fk~ 1. If n is a
Fibonacci number, (6.113) holds with r = 1 and kl = k. Otherwise n - Fk
has a Fibonacci representation FkL +. + Fk,-, by induction on n; and (6.113)
holds if we set kl = k, because the inequalities FkL < n - Fk < Fk 1 imply
that k > kz.) Conversely, any representation of the form (6.113) implies that
h, < n < h,+l ,
because the largest possible value of FkJ + . . . + Fk, when k >> kz >> . . . >>
k, >> 0 is
Fk~2$.Fk~4+...+FkmodZf2 = Fk~m, -1, if k 3 2. (6.114)
(This formula is easy to prove by induction on k; the left-hand side is zero
when k is 2 or 3.) Therefore k1 is the greedily chosen value described earlier,
and the representation must. be unique.
Any unique system of representation is a number system; therefore Zeckendorf’s
theorem leads to the Fibonacci number system. We can represent
any nonnegative integer n as a sequence of O’s and 1 ‘s, writing
n = (b,b,-1 . ..bl)F w n = bkhc . (6.115)
This number system is something like binary (radix 2) notation, except that
there never are two adjacent 1's. For example, here are the numbers from 1
to 20, expressed Fibonacci-wise:
1 = (000001)~ 6 = (OOIOO1)F 11 = (010100)~ 16 = (lOOIOO)F
2 = (000010)~ 7 = (001010)~ 12 = (010101)~ 17= (100101)~
3 = (000100)~ 8 = (OIOOOO)F 13= (100000)~ 18 = (lOIOOO)F
4 = (000101)~ 9 = (010001)~ 14= (100001)~ 19 = (101001)~
5 = (001000)~ 10 = (010010)~ 15= (100010)~ 20 = (101010)~
The Fibonacci representation of a million, shown a minute ago, can be contrasted
with its binary representation 219 + 218 + 2” + 216 + 214 + 29 + 26:
(1000000)10 = (10001010000000000010100000000)~
k=2
= (11110100001001000000)~.
The Fibonacci representation needs a few more bits because adjacent l's are
not permitted; but the two representations are analogous.
To add 1 in the Fibonacci number system, there are two cases: If the
“units digit” is 0, we change it to 1; that adds F2 = 1, since the units digit