Look at the sequence of differences finding that it is a geometric sequence with common ratio ##2## and hence derive the recursive formula:

##a_1 = 3##

##a_(n+1) = 2a_n + 1##

Write out the original sequence:

##37153163127##

Write out the sequence of differences of that sequence:

##48163264##

This is a geometric sequence with common ratio ##2##.

Try subtracting it from the original sequence to find:

##-1-1-1-1-1##

So we can deduce the recursive rule:

##a_1 = 3##

##a_(n+1) = 2(a_n + 1) – 1 = 2a_n+1##

A direct expression for ##a_n## is:

##a_n = 2^(n+1)-1##