Download 1. Find a linear recurrence for the sequence 0, 1, 3, 7, 15, 31, 63, 127

1. Find a linear recurrence for the sequence
(an = 2n − 1).
0, 1, 3, 7, 15, 31, 63, 127, . . .
2. Find a linear recurrence for the sequence
0, 1, 1, 4, 9, 25, 64, 169, 441, . . .
(an = Fn2 ).
3. Let a0 = a1 = 1 and
an+2 = 2an+1 + an ,
n = 0, 1, 2, . . . .
Find an explicit formula for an , and find the value of the limit
an+1
.
n→∞ an
lim
Bonus Problem 1. Find the values of the infinite series
∞
X
Fn
1 1 2
3
5
8
13
= + + +
+
+
+
+ ...
n
2
2
4
8
16
32
64
128
n=1
and
∞
X
Fn
1 2
3
5
8
13
=1+ + +
+
+
+
+ ....
n!
2
6
24
120
720
5040
n=1
Bonus Problem 2. Let p ≥ 1 be an integer, and let ω = e2πi/p . Prove that any sequence
of complex numbers a0 , a1 , a2 , . . . that is periodic with period p (that is,
an+p = an for all n ≥ 0) can be written in the form
an = c0 + c1 ω n + c2 ω 2n + . . . + cp−1 ω (p−1)n
for some constants c0 , . . . , cp−1 ∈ C.
Bonus Problem 3. If an satisfies a linear recurrence of order k, and bn satisfies a linear recurrence of order `, show that
• an + bn satisfies a linear recurrence of order k + `.
• an bn satisfies a linear recurrence of order k`.
• a2n satisfies a linear recurrence of order k+1
2 .
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