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31
2007 Guts #31: Recursion without Starting Value
2007 Guts #31: Recursion without Starting Value
Source:
June 22, 2012
Problem Statement
A sequence
{
a
n
}
n
≥
0
\{a_n\}_{n\geq 0}
{
a
n
}
n
≥
0
of real numbers satisfies the recursion
a
n
+
1
=
a
n
3
−
3
a
n
2
+
3
a_{n+1}=a_n^3-3a_n^2+3
a
n
+
1
=
a
n
3
−
3
a
n
2
+
3
for all positive integers
n
n
n
. For how many values of
a
0
a_0
a
0
does
a
2007
=
a
0
a_{2007}=a_0
a
2007
=
a
0
?
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