# Some day in the park with Fiqo

“Awwwwhat a cute little diagram,” the stranger said, “what’s its name?”

“Fi qo.   (A misspelling; it isn’t purebred.)”

“May I pet it?”

“Sure, Fiqo won’t bite, but it can have a bit of a bad attitude.”

“?”

“As if it’s trying to prove something.”

“Just what is Fiqo trying to prove?

The answer is a mathematical formula not meant to be tricky, though the general-case role of those black squares (unit sized in this specific case) might take an extra moment to understand precisely.

I think that Fiqo is trying to prove that:

For any sequence $a_1,a_2,a_3,...$ where $\forall n\ge1, a_n+a_{n+1}=a_{n+2}$, $a_n=a_2+a_1+a_2+a_3+a_4+...+a_{n-2}$ (these are Fibonacci-type sequences).

Like so:

The layers for 5 and 4 haven't been done because they are too small. (Sorry to those who are colourblind)

The reason Fiqo is not purebred is:

Because this isn't a Fibonacci sequence - $a_1=3$ instead of $1$. I suppose it would be called Fibo instead, and this is a play on the common dog name Fido.

• Thank you for mentioning how Fiqo was named. Did you have this formula ready to go or figure it out from the picture? Would be nice to describe how it relates in any case. – humn Jan 13 '17 at 12:41
• I knew of it but had to derive it from the picture, @humn. – boboquack Jan 13 '17 at 23:42
• You made an explanation-without-words for this proof-without-words. Just perfect! The bounty will take a week to work its way through the system. – humn Jan 14 '17 at 0:02
• Check, boboquack, as in checkmark – humn Jan 14 '17 at 6:54