MathDB

In the dog dictionary the words are any sequence of letters

A

and

U

for example

AA

UAU

and

AUAU

. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters. For each positive integer

n

, determine the largest "profundity" of word, in dog dictionary, can have with

n

letters. Note: The word

AAUUA

has "profundity"

14

because your subwords are

A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA

Problem Statement