A Formula for a Staircase

★ 5/10number theory sequences Estimation Casework

Problem

Consider the sequence $\{a_n\}$ :

1,\; 3, 3, 3,\; 5, 5, 5, 5, 5,\; \dots

in which each positive odd number $k$ appears exactly $k$ times. Do there exist integers $r, s, t$ such that

a_n = r \cdot \left\lfloor \sqrt{n + s} \right\rfloor + t

for every positive integer $n$ ? (Here $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$ .)

Answer

Solution

Difficulty	5/10
Topics	number theory, sequences, Estimation, Casework

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