The
th number in the sequence can be expressed as

Extracting the
th term from the sum gives

and
. 3 divides both 12 and 21, so
and 21 contribute no remainder.
This leaves us with

Recall that a decimal integer is divisible by 3 if its digits add to a multiple of 3. The digits in
are
copies of 2 and one 0, so the digital sum is
.
- If
for
, then the digital sum is
, which is not divisible by 3. - If
, then the sum is
, which is not divisible by 3. - If
, then the sum is
, which is always divisble by 3.
This means that roughly 1/3 of the first
numbers in this sequence are divisible by 3; among the first 100 terms, they occur for
, of which there are 33.