Concrete mathematics : a foundation for computer science

6.7 CONTINUANTS 289
We also know that the total number of terms in a continuant is a Fibonacci
number; hence we have the identity
F,,+I = 2 (“; “)
k=O
(6.130)
(A closed form for (6.12g), generalizing the Euler-Binet formula (6.123) for
Fibonacci numbers, appears in (5.74).)
The relation between continuant polynomials and Morse code sequences
shows that continuants have a mirror symmetry:
K(x,, . . . , x2,x1) = K(x1,xr,...,xn). (6.131)
Therefore they obey a recurrence that adjusts parameters at the left, in addition
to the right-adjusting recurrence in definition (6.127):
K,(xI,... ,%I) = XI& 1(X2,...,&1) +Kn 2(x3,...,&). (6.132)
Both of these recurrences are special cases of a more general law:
K m+*(X1,...,X,,X,+1,~..,x~+~)
= K,(xl,...,x,)K,(x,+~,...,x,+,)
+kn I(xI,...,x, l)K, 1(~,+2,...,~rn+n). (6.133)
This law is easily understood from the Morse code analogy: The first product
K,K, yields the terms of K,+, in which there is no dash in the [m, m + 11
position, while the second product yields the terms in which there is a dash
there. If we set all the x’s equal to 1, this identity tells us that Fm+n+l =
Fm+lF,+l + F,F,; thus, (6.108) is a special case of (6.133).
Euler [90] discovered that continuants obey an even more remarkable law,
which generalizes Cassini’s identity:
K m+n(Xlr.~~ t Xm+n) Kk(Xm+l, . . . , %n+k)
= kn+k(Xl, . . . rX,+k)K,(x,+l,...,x,+,)
+ (-l)kKm I(XI,...,X, l)Kn k 1(%n+k+2,...,Xm+,). (6.134)
This law (proved in exercise 29) holds whenever the subscripts on the K’s are
all nonnegative. For example, when k = 2, m = 1, and n = 3, we have
K(xl,x2,x3,x4)K(x2,x3) = K(Xl,X2,X?,)K(XL,X3,X4) +1
Continuant polynomials are intimately connected with Euclid’s algorithm.
Suppose, for example, that the computation of gcd(m, n) finishes