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One sequence problem

Source: 2019 CSMO Grade 11 P5

July 31, 2019
algebraSequence

Problem Statement

For positive integer n, define ana_n as the number of the triangles with integer length of every side and the length of the longest side being 2n.2n. (1) Find ana_n in terms of n;n; (2)If the sequence {bn}\{ b_n\} satisfying for any positive integer n,n, ∑k=1n(−1)n−k(nk)bk=an.\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n. Find the number of positive integer nn satisfying that bn≤2019an.b_n\leq 2019a_n.
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