Concrete mathematics : a foundation for computer science

300 SPECIAL NUMBERS
44 Prove the converse of Cassini’s identity (6.106): If k and m are integers
such that Im2-km-k21 = 1, then there is an integer n such that k = fF,
and m = fF,+l.
45 Use the repertoire method to solve the general recurrence
X0 = a; x, = p; Xn = X,--l +X,-2+yn+6.
46 What are cos 36” and cos 72”?
47 Show that
2"~'h = ; (2;,)5k,
and use this identity to deduce the values of F, mod p and F,+1 mod p
when p is prime.
48 Prove that zero-valued parameters can be removed from continuant polynomials
by collapsing their neighbors together:
K,(xl,... ,xTl-1,0,x m+l,...,Xn)
= K,-2(x,,. . . , Xm~Z,Xm~l+X,+l,X,+Z,...,X,),
l 0 such that a,, is divisible by 5.