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PEN M Problems
2
M 2
M 2
Source:
May 25, 2007
floor function
induction
Recursive Sequences
Problem Statement
An integer sequence
{
a
n
}
n
≥
1
\{a_{n}\}_{n \ge 1}
{
a
n
}
n
≥
1
is defined by
a
1
=
1
,
a
n
+
1
=
a
n
+
⌊
a
n
⌋
.
a_{1}=1, \; a_{n+1}=a_{n}+\lfloor \sqrt{a_{n}}\rfloor.
a
1
=
1
,
a
n
+
1
=
a
n
+
⌊
a
n
⌋
.
Show that
a
n
a_{n}
a
n
is a square if and only if
n
=
2
k
+
k
−
2
n=2^{k}+k-2
n
=
2
k
+
k
−
2
for some
k
∈
N
k \in \mathbb{N}
k
∈
N
.
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