MathDB

For a positive integer

n

, an

n

-sequence is a sequence

(a_0,\ldots,a_n)

of non-negative integers satisfying the following condition: if

i

and

j

are non-negative integers with

i+j \leqslant n

, then

a_i+a_j \leqslant n

and

a_{a_i+a_j}=a_{i+j}

.
Let

f(n)

be the number of

n

-sequences. Prove that there exist positive real numbers

c_1

c_2

, and

\lambda

such that

c_1\lambda^n<f(n)<c_2\lambda^n

for all positive integers

n

A8