'''There's a difference between adding an <math>n-1^{th}</math> unknown digit after <math>n-2</math> known digits, and adding <math>n-2</math> unknown digits before one known digit in the <math>n-1</math> possition.
'''We apply the first one when expressing <math>f(n-1)</math> - that is, if <math>n-2</math> positions are know, then if the <math>n-2</math> position is odd, indeed you have <math>10</math> ways to fill the <math>n-1^{th}</math> position , and if it's even you have 5 ways.
'''We apply the second one when expressing <math>f^{(odd)}(n-1)</math> - that is, if the <math>n-1</math> position is known and odd then you have <math>f(n-2)</math> ways to fill in the <math>n-2</math> previous positions.
'''Adi