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b_n = \sum\limits_{r+s \leq n} a_ra_s if a_n = 2^{n-1}

Source: 1974 Swedish Mathematical Competition p1

March 26, 2021
SequencealgebraSum

Problem Statement

Let an=2n1a_n = 2^{n-1} for n>0n > 0. Let bn=r+snaras b_n = \sum\limits_{r+s \leq n} a_ra_s Find bnbn1b_n-b_{n-1}, bn2bn1b_n-2b_{n-1} and bnb_n.