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TOT 348 1992 Autumn A J6 a(n + 1) = a(n) + [\sqrt{a(n)}]

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June 10, 2024
algebranumber theoryrecurrence relationfloor functionPerfect Square

Problem Statement

Consider the sequence a(n)a(n) defined by the following conditions: a(1)=1    a(n+1)=a(n)+[a(n)]   ,    n=1,2,3,...a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,... Prove that the sequence contains an infinite number of perfect squares. (Note: [x][x] means the integer part of xx, that is the greatest integer not greater than xx.)
(A Andjans)
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