Concrete mathematics : a foundation for computer science

Well, y must be
irrational, because
of a little-known
Einsteinian assertion:
“God does
not throw huge
denominators at
the universe.”
6.7 CONTINUANTS 293
Therefore nobody believes that y is rational; but nobody so far has been able
to prove that it isn’t.
Let’s conclude this chapter by proving a remarkable identity that ties a lot
of these ideas together. We introduced the notion of spectrum in Chapter 3;
the spectrum of OL is the multiset of numbers Ln&], where 01 is a given constant.
The infinite series
can therefore be said to be the generating function for the spectrum of @,
where @ = (1 + fi)/2 is the golden ratio. The identity we will prove, discovered
in 1976 by J.L. Davison [61], is an infinite continued fraction that
relates this generating function to the Fibonacci sequence:
(6.143)
Both sides of (6.143) are interesting; let’s look first at the numbers Ln@J.
If the Fibonacci representation (6.113) of n is Fk, + . . . + Fk,, we expect n+
to be approximately Fk, +I +. . . + Fk,+i , the number we get from shifting the
Fibonacci representation left (as when converting from miles to kilometers).
In fact, we know from (6.125) that
n+ = Fk,+, + . . . + Fk,+l - ($“I + . + q”r) .
Now+=-l/@andki >...>>k,>>O,sowehave
and qkl +.. .+$jkl has the same sign as (-1) kr, by a similar argument. Hence
In+] = Fk,+i +.‘.+Fk,+l - [k,(n) iseven]. (6.144)
Let us say that a number n is Fibonacci odd (or F-odd for short) if its least
significant Fibonacci bit is 1; this is the same as saying that k,(n) = 2.
Otherwise n is Fibonacci even (F-even). For example, the smallest F-odd