the formula is f( n + 1 ) = f( n ) + 2*f( n - 1 ) + 1

and you are making mistake because you did:

f( n + 1 ) = f( n ) +2*( n - 1 ) + 1

answers:

f( 2 ) = f( 1 ) + 2*f( 1 - 1 ) + 1 = 1 + 2*0 + 1 = 2

f( 3 ) = f( 2 ) + 2*f( 1 ) + 1 = 2 + 2*1 + 1 = 5

f( 4 ) = f( 3 ) + 2*f( 2 ) + 1 = 5 + 2*2 + 1 = 10

f( 5 ) = f( 4 ) + 2*f( 3 ) + 1 = 10 + 2*5 + 1 = 21