Variation on a theme
Instead of adding the last two numbers together (as in the standard Fibonacci numbers), let's give ourselves a choice:
either add the current number to its parent or to its grandparent:
The PseudoFibonacci numbers^{1} are formed by starting with
and thereafter, forming the next number in
sequence with a choice:

You can explore, yourself, a bit, by clicking on the buttons below:
For more about these, take a look here
^{1} Relatives of the Fibonacci numbers, these are also known as "Dailey numbers," after their discoverer.