Advent calendar 2020

To make sequences of 4 numbers, we can insert 4s into three different places in the length 3 sequences given to obtain:

There are some possibilities missing: those containing $i,4,i+1$. These can be found by taking the sequences of length 2, picking a number $i$, adding 1 to every number larger than $i$, then replacing $i$ with $i4i+1$.

This gives a total of 3×3+2×1=11 sequences for 4 numbers.

To make sequences with 5 numbers, we can insert 5s into four different places in the length 4 sequences. This gives 4×11=44 sequences.

The missing sequences can then be found by taking the sequences of length 3, then doing the same process as above:

There are 3×3=9 of these, giving 44+9 = 53 total sequences of length 5.

To make sequences with 6 numbers, we can insert 6s into five different places in the length 5 sequences. This gives 5×53=265 sequences. We can also make sequence by picking a number to replace in the length 4 sequences. This gives 4×11=44 more sequences. Therefore there are 265+44 = 309 sequences in total.