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 3 of 5: Use divisibility between sharp bounds

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(n^2+5n+3)a_n<a_{n+2}<(n^2+5n+5)a_n\Longrightarrow a_{n+2}=(n+1)(n+4)a_n\quad(n\ge4)

Detailed analysis

Substitute the recurrence twice and use the lower bound to obtain the strict inequalities. Since $a_n$ divides $a_{n+2}$ , the quotient is an integer strictly between two consecutive possible integer coefficients; it must be $n^2+5n+4$ , which factors as shown.