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Vojtěch Jarník IMC
1996 VJIMC
Problem 2
binomial sum with recurrence relation
binomial sum with recurrence relation
Source: VJIMC 1996 1.2
October 12, 2021
Sequences
recurrence relation
Summation
Binomial
algebra
Problem Statement
Let
{
a
n
}
n
=
0
∞
\{a_n\}^\infty_{n=0}
{
a
n
}
n
=
0
∞
be the sequence of integers such that
a
0
=
1
a_0=1
a
0
=
1
,
a
1
=
1
a_1=1
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
−
2
a
n
a_{n+2}=2a_{n+1}-2a_n
a
n
+
2
=
2
a
n
+
1
−
2
a
n
. Decide whether
a
n
=
∑
k
=
0
⌊
n
2
⌋
(
n
2
k
)
(
−
1
)
k
.
a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.
a
n
=
k
=
0
∑
⌊
2
n
⌋
(
2
k
n
)
(
−
1
)
k
.
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