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Korea Contests
Korea National Olympiad
1997 Korea National Olympiad
2
Classic Sum
Classic Sum
Source: 1997 Korean National Olympiad #2
March 18, 2018
algebra
combinatorics
Problem Statement
For positive integer
n
,
n,
n
,
let
a
n
=
∑
k
=
0
[
n
2
]
(
n
−
2
k
)
(
−
1
4
)
k
.
a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.
a
n
=
∑
k
=
0
[
2
n
]
(
k
n
−
2
)
(
−
4
1
)
k
.
Find
a
1997
.
a_{1997}.
a
1997
.
(For real
x
,
x,
x
,
[
x
]
[x]
[
x
]
is defined as largest integer that does not exceeds
x
.
x.
x
.
)
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