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A finite sequence

x_1,\dots,x_r

of positive integers is a palindrome if

x_i=x_{r+1-i}

for all integers

1 \le i \le r

. Let

a_1,a_2,\dots

be an infinite sequence of positive integers. For a positive integer

j \ge 2

, denote by

a[j]

the finite subsequence

a_1,a_2,\dots,a_{j-1}

. Suppose that there exists a strictly increasing infinite sequence

b_1,b_2,\dots

of positive integers such that for every positive integer

n

, the subsequence

a[b_n]

is a palindrome and

b_{n+2} \le b_{n+1}+b_n

. Prove that there exists a positive integer

T

such that

a_i=a_{i+T}

for every positive integer

i

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