Problem 5

Determine all sequences

a_0,a_1,a_2,\ldots

of positive integers with

a_0\ge2015

such that for all integers

n\ge1

: (i)

a_{n+2}

is divisible by

a_n

; (ii)

|s_{n+1}-(n+1)a_n|=1

, where

s_{n+1}=a_{n+1}-a_n+a_{n-1}-\cdots+(-1)^{n+1}a_0

Step 2 of 5: Obtain a uniform lower bound

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a_n\ge223\quad(n\ge0)

Detailed analysis

The case n=1 gives a_0=3a_1-a_2 plus or minus one, so the large initial value forces a_1 at least 671. The case n=2 together with divisibility by a_1 then forces a_2 at least 223. The recurrence propagates this lower bound to every term.