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TOT 354 1992 Autumn A S3 a(n + 1) = a(n) + [\sqrt{a(n)}]

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June 10, 2024
number theoryPerfect Squarerecurrence relationfloor function

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,... How many perfect squares no greater in value than 10000001000 000 will be found among the first terms of the sequence? ( (Note: [x][x] means the integer part of xx, that is the greatest integer not greater than xx.)
(A Andjans)
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