Q:

Find the combination algorithm

Once a month, Jerry visits the restaurant that he owns. Being paranoid that someone will steal his fortune, on each visit he changes the combination to his personal safe, which is hidden in the back part of the managers' office.

On each update, Jerry uses the next number in a numerical pattern that he has been aware of for years, and is very confident that no one else will figure out. Each time, he jots the previous combination down on a sheet of paper, which is taped to the side of the safe, to remind him where he is in the sequence.

The safe has a simple, 10-button keypad, like below, and can be programmed to accept combinations from length 1 to length 15. After 3 failed attempts in a row to open the safe, a text will be sent to Jerry's phone. Therefore, guessing is not a great idea.

After some time, the restaurant managers find the small piece of paper taped to the safe, and figure out that the numbers are the previous combinations, and assume that a pattern or sequence of some kind is being followed. They are bored and would love to break into the safe, if only for the challenge. However, they have yet to be able to find the pattern.
Here is the list of previous combinations:

1
  8
  41
  208
  1041
  5208
  26041
  130208
  651041
  3255208
  16276041
  81380208
  406901041

What is the current combination?
Hint #1:

 Eventually, Jerry's algorithm will produce combinations that are too long (>15 digits), but for now this works.

Hint #2

 The sequence has nothing to do with letters or alphabets or changing bases, and nothing is encrypted.

A:

To arrive at the new combination,

 Jerry takes the previous combination, multiplies it by $5$, then adds either $1$ or $3$, depending on whether the previous combination was even or odd (respectively).

The next combination is

 $406901041\times 5+3=2034505208$.

