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Part of 2019 Silk Road

Problems(1)

combinatorics with floor function

Source: SRMC 2019 P4

7/16/2019
The sequence {an} \{a_n \} is defined as follows: a0=1 a_0 = 1 and an=∑k=1[n]an−k2 {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} for n≥1. n \ge 1. Prove that among a1,a2,…,a106 a_1, a_2, \ldots, a_ {10 ^ 6} there are at least 500500 even numbers. (Here, [x] [x] is the largest integer not exceeding x x .)
combinatoricsSumfloor functionEvenalgebrafunction