MathLabs

Problem 5

Determine all sequences a0,a1,a2,…a_0,a_1,a_2,\ldots of positive integers with a0≥2015a_0\ge2015 such that for all integers n≥1n\ge1: (i) an+2a_{n+2} is divisible by ana_n; (ii) ∣sn+1−(n+1)an∣=1|s_{n+1}-(n+1)a_n|=1, where sn+1=an+1−an+an−1−⋯+(−1)n+1a0s_{n+1}=a_{n+1}-a_n+a_{n-1}-\cdots+(-1)^{n+1}a_0.
Step 1 of 5: Convert the absolute value into a recurrence
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sn+1=(n+1)an+hn,hn∈{−1,1},an+1=(n+1)an+nan−1+δn,δn∈{−2,0,2}s_{n+1}=(n+1)a_n+h_n,\quad h_n\in\{-1,1\},\qquad a_{n+1}=(n+1)a_n+na_{n-1}+\delta_n,\quad \delta_n\in\{-2,0,2\}
Detailed analysis

Introduce the signs h_n. Comparing the definitions of two consecutive alternating sums and using the identity that their sum is the next term gives the displayed recurrence, with an error term restricted to three even values.