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The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Tutorial 11 Week 12
1.
Find a closed form for the generating function of the sequence
a
0
, a
1
, . . . ,
where
a
0
= 1
, a
1
= 1 and
a
n
= 5
a
n

1

6
a
n

2
.
2.
Let
a
0
,
a
1
,
a
2
,
. . .
, be a sequence of integers satisfying the recurrence relation
a
n

2
a
n

1

a
n

2
+ 2
a
n

3
= 0
,
for
n
≥
4, where
a
0
= 0,
a
1
= 0,
a
2
= 2 and
a
3
= 5. Find a closed form for the
generating function of this sequence.
3.
Solve the following recurrence relations using generating functions:
(
i
)
x
n
= 4
x
n

1

3
x
n

2
, where
x
0
= 1 and
x
1
= 2.
(
ii
)
x
n
= 3
x
n

1

3
x
n

2
+
x
n

3
, where
x
0
= 0,
x
1
= 1 and
x
2
= 3.
4.
Consider the recurrence relation
a
n
=
a
n

1
+ 4
,
for
n >
0 and where
a
0
= 1
.
Find a closed form for its generating function and
thereby ±nd a formula for
a
n
.
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View Full Document The University of Sydney
MATH 1004
Second Semester
Discrete Mathematics
2011
Problem Set 11
1.
For each of the following recurrence relations, ±nd a closed form for its gener
ating function:
(
i
)
x
n
+2

5
x
n
+1
+ 6
x
n
= 0 for all
n
≥
0, with
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This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.
 Three '11
 various

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