Concrete mathematics : a foundation for computer science

298 SPECIAL NUMBERS
24 Prove that the tangent number Tz,+l is a multiple of 2”. Hint: Prove
that all coefficients of Tz,,(x) and Tzn+l (x) are multiples of 2”.
25 Equation (6.57) proves that the worm will eventually reach the end of
the rubber band at some time N. Therefore there must come a first
time n when he’s closer to the end after n minutes than he was after
n - 1 minutes. Show that n < :N.
26 Use summation by parts to evaluate S, = xr=, Hk/k. Hint: Consider
also the related sum Et=, Hk-r/k.
2’7 Prove the gcd law (6.111) for Fibonacci numbers.
28 The Lucas number L, is defined to be Fn+r + F,--r. Thus, according to
(6.log), we have Fzn = F,L,. Here is a table of the first few values:
nl 0 1 2 3 4 5 6 7 8 9 10 11 12 13
L,,I 2 1 3 4 7 11 18 29 47 76 123 199 322 521
a Use the repertoire method to show that the solution Qn to the general
recurrence
Qo = a; Ql = B; Qn = Qn-l+Qn-2, n>l
can be expressed in terms of F, and L,.
b Find a closed form for L, in terms of 4 and $.
29 Prove Euler’s identity for continuants, equation (6.134).
30 Generalize (6.136) to find an expression for the incremented continuant
K(x,, . . . ,~,,~l,~~+y,~~+l,..., x,,), when 16 m