2021 AMC 12A Spring Problem 8

Question

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?

(O, E, O)

(E, E, O)

(E, O, E)

(O, O, E)

(O, O, O)

Only parities matter, so compute the sequence with each term as

E

or

O

using

O+O=E

The parity sequence repeats with period

7;

reduce

2021, 2022, 2023

modulo

7

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

as

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.

Problem 8 in other years