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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2000 Moldova Team Selection Test
9
9
Part of
2000 Moldova Team Selection Test
Problems
(1)
Sequence
Source:
2/21/2016
The sequence
x
n
x_{n}
x
n
is de fined by:
x
0
=
1
,
x
1
=
0
,
x
2
=
1
,
x
3
=
1
,
x
n
+
3
=
(
n
2
+
n
+
1
)
(
n
+
1
)
n
x
n
+
2
+
(
n
2
+
n
+
1
)
x
n
+
1
−
n
+
1
n
x
n
(
n
=
1
,
2
,
3..
)
x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)
x
0
=
1
,
x
1
=
0
,
x
2
=
1
,
x
3
=
1
,
x
n
+
3
=
n
(
n
2
+
n
+
1
)
(
n
+
1
)
x
n
+
2
+
(
n
2
+
n
+
1
)
x
n
+
1
−
n
n
+
1
x
n
(
n
=
1
,
2
,
3..
)
Prove that all members of the sequence are perfect squares.
Sequence
Perfect Squares
Integers
number theory with sequences