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Korea Winter Program Practice Test
2016 Korea Winter Program Practice Test
1
Find all {a_n}
Find all {a_n}
Source: 2016 Korea Winter Program Test1 Day2 #5
January 25, 2016
algebra
algebra proposed
Integer sequence
number theory
Problem Statement
Find all
{
a
n
}
n
≥
0
\{a_n\}_{n\ge 0}
{
a
n
}
n
≥
0
that satisfies the following conditions.(1)
a
n
∈
Z
a_n\in \mathbb{Z}
a
n
∈
Z
(2)
a
0
=
0
,
a
1
=
1
a_0=0, a_1=1
a
0
=
0
,
a
1
=
1
(3) For infinitly many
m
m
m
,
a
m
=
m
a_m=m
a
m
=
m
(4) For every
n
≥
2
n\ge2
n
≥
2
,
{
2
a
i
−
a
i
−
1
∣
i
=
1
,
2
,
3
,
⋯
,
n
}
≡
{
0
,
1
,
2
,
⋯
,
n
−
1
}
\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}
{
2
a
i
−
a
i
−
1
∣
i
=
1
,
2
,
3
,
⋯
,
n
}
≡
{
0
,
1
,
2
,
⋯
,
n
−
1
}
m
o
d
n
\mod n
mod
n
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