for the rest of the array we will rely on the previous entries to compute new ones
for ( $ N = 1 ; $ N <= 20 ; $ N += 1 ) echo ( " Maximum ▁ Number ▁ of ▁ A ' s ▁ with " . $ N . " keystrokes ▁ is ▁ " . findoptimal ( $ N ) . " STRNEWLINE " ) ; ? >