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Part of 2007 Cuba MO

Problems(1)

a_{n+1} = S(a_n) is periodic, where S(n) = sum of digits of n^2 +1

Source: 2007 Cuba MO 2.8

9/15/2024
For each positive integer nn, let S(n)S(n) be the sum of the digits of n2+1n^2 +1. A sequence {an}\{a_n\} is defined, with a0a_0 an arbitrary positive integer and an+1=S(an)a_{n+1} = S(a_n). Prove that the sequence {an}\{a_n\} is eventually periodic with period three.
number theorysum of digitsPeriodic sequenceperiodic