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PEN L Problems
1
L 1
L 1
Source:
May 25, 2007
induction
algebra
binomial theorem
Linear Recurrences
pen
Problem Statement
An integer sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
+
a
n
a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}
a
0
=
0
,
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
+
a
n
Show that
2
k
2^{k}
2
k
divides
a
n
a_{n}
a
n
if and only if
2
k
2^{k}
2
k
divides
n
n
n
.
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