MathDB

The infinite sequence

a_0,a _1, a_2, \dots

of (not necessarily distinct) integers has the following properties:

0\le a_i \le i

for all integers

i\ge 0

, and

\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k

for all integers

k\ge 0

. Prove that all integers

N\ge 0

occur in the sequence (that is, for all

N\ge 0

, there exists

i\ge 0

with

a_i=N

Problem Statement