MathDB

Consider sequences

a_0

a_1

a_2

\cdots

of non-negative integers defined by selecting any

a_0

a_1

a_2

(not all 0) and for each

n

\geq

3 letting

a_n

= |

a_n-1

a_n-3

|
1-In the particular case that

a_0

= 1,

a_1

= 3 and

a_2

= 2, calculate the beginning of the sequence, listing

a_0

a_1

\cdots

a_{19}

a_{20}

.
2-Prove that for each sequence, there is a constant

c

such that

a_i

\leq

c

for all

i

\geq

0. Note that the constant

c

my depend on the numbers

a_0

a_1

and

a_2

3-Prove that, for each choice of

a_0

a_1

and

a_2

, the resulting sequence is eventually periodic.
4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values

a_0

a_1

a_2

of the sequence

Problem Statement