MathDB

Define a sequence

(a_n)

a_1 =1, a_2 =2,

and

a_{k+2}=2a_{k+1}+a_k

for all positive integers

k

. Determine all real numbers

\beta >0

which satisfy the following conditions:
(A) There are infinitely pairs of positive integers

(p,q)

such that

\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.

(B) There are only finitely many pairs of positive integers

(p,q)

with

\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }

for which there is no index

k

with

q=a_k.

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