MathDB

Call a positive integer

n\geq 2

junk if there exist two distinct

n

digit binary strings

a_1a_2\cdots a_n

and

b_1b_2\cdots b_n

such that
[*]

a_1+a_2=b_1+b_2,

[*]

a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}

for all

2\leq i\leq n-1,

and [*]

a_{n-1}+a_n=b_{n-1}+b_n

.
Find the number of junk positive integers less than or equal to

2016

.
Proposed by Nathan Ramesh

Problem Statement