Problem 191

Prize Strings

A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.

During an n-day period a trinary string is formed for each child consisting of L’s (late), O’s (on time), and A’s (absent).

Although there are eighty-one trinary strings for a 4-day period that can be formed, exactly forty-three strings would lead to a prize:

OOOO OOOA OOOL OOAO OOAA OOAL OOLO OOLA OAOO OAOA
OAOL OAAO OAAL OALO OALA OLOO OLOA OLAO OLAA AOOO
AOOA AOOL AOAO AOAA AOAL AOLO AOLA AAOO AAOA AAOL
AALO AALA ALOO ALOA ALAO ALAA LOOO LOOA LOAO LOAA
LAOO LAOA LAAO

How many “prize” strings exist over a 30-day period?

出勤奖励

某所学校给有出勤和守时表现良好的孩子发放现金奖励。如果孩子连续三天缺席，或是有多于一次迟到，则拿不到这份奖励。

根据n天的实际出勤情况，我们可以生成L（迟到）、O（准时）和A（缺席）这三个字母组成的字符串。

根据4天的出勤情况，能够生成的字符串一共有81种可能，其中恰好有43个串可以获得奖励：

根据30天的出勤情况生成的字符串中，有多少个是可以获得奖励的串？