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Putnam
1999 Putnam
3
Putnam 1999 A3
Putnam 1999 A3
Source:
December 22, 2012
Putnam
induction
college contests
Problem Statement
Consider the power series expansion
1
1
−
2
x
−
x
2
=
∑
n
=
0
∞
a
n
x
n
.
\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.
1
−
2
x
−
x
2
1
=
n
=
0
∑
∞
a
n
x
n
.
Prove that, for each integer
n
≥
0
n\geq 0
n
≥
0
, there is an integer
m
m
m
such that
a
n
2
+
a
n
+
1
2
=
a
m
.
a_n^2+a_{n+1}^2=a_m.
a
n
2
+
a
n
+
1
2
=
a
m
.
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