2021 AMC 12A Problem 8

Problem 8 of 25EasierAlgebra Number Theory

A sequence of numbers is defined by

D_0 = 0,

D_1 = 0,

D_2 = 1,

and

D_n = D_{n-1} + D_{n-3}

for

n \ge 3.

What are the parities (evenness or oddness) of the triple of numbers

(D_{2021}, D_{2022}, D_{2023}),

where

E

denotes even and

O

denotes odd?

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Solution

Working modulo

2,

the terms

D_0, D_1, D_2, \ldots

have parities

E,

E,

O,

O,

O,

E,

O,

E,

E,

O,

O,

O,

E,

O, \ldots

which repeat with period

7

starting from

D_0

(indeed

D_7, D_8, D_9

have the same parities

E, E, O

D_0, D_1, D_2

). Since

2021 \equiv 5,

2022 \equiv 6,

and

2023 \equiv 0 \pmod 7,

the parities match those of

D_5, D_6, D_0,

namely

E, O, E.

Thus, the correct answer is C.