Fibonachos

A few weeks ago I was going through my saved tweets, and I saw this one from Math Lady Hazel which reminded me of a story from early 2017 that I played a small part in.

In January 2017, reddit user Teblefer asked the following question:

Two people are sharing a plate of nachos. They take turns dividing the nachos, each taking the nth Fibonacci number of nachos on the nth turn. When the number of nachos left is less than the next Fibonacci number, they start the sequence over. What number of nachos (less than 500) requires the most number of restarts? How would you generate numbers of nachos with a high number of restarts?

Fibonachos
byu/Teblefer inmath

I computed the first few terms, added them to the OEIS as A280521 and A280523, and wrote:

I’m shocked that I didn’t find this sequence in the OEIS!

a(1)  = 1 via [1]
a(2)  = 1 via [1, 1]
a(3)  = 2 via [1, 1], [1]
a(4)  = 1 via [1, 1, 2]
a(5)  = 2 via [1, 1, 2], [1]
a(6)  = 2 via [1, 1, 2], [1, 1]
a(7)  = 1 via [1, 1, 2, 3]
a(8)  = 2 via [1, 1, 2, 3], [1]
a(9)  = 2 via [1, 1, 2, 3], [1, 1]
a(10) = 3 via [1, 1, 2, 3], [1, 1], [1]
a(11) = 2 via [1, 1, 2, 3], [1, 1, 2]
a(12) = 1 via [1, 1, 2, 3, 5]

Here are where records appear (I also didn’t find this in the OEIS):

a(1)    = 1
a(3)    = 2
a(10)   = 3
a(30)   = 4
a(84)   = 5
a(227)  = 6
a(603)  = 7
a(1589) = 8
a(4172) = 9

Conjecture: Records appear at A215004(2n).

A215004(0) = A215004(1) = 1; for n>1, A215004(n) = A215004(n-2) + A215004(n-1) + floor(n/2).

Neil Sloane’s Winter Fruits

Later that month on January 26, 2017, Neil Sloane (the founder of the OEIS) gave a talk in Doron Zeilberger’s Experimental Mathematics Seminar at Rutgers titled “Winter Fruits: New Problems from OEIS December 2016–January 2017.” You can see a PDF of the slides here.

The lecture was recorded, and when I first got to the 12:50 minute mark, I was surprised to see my name appear on the screen.

Other nacho sequences

You can play the “Fibonachos” game for other sequences too, and Neil’s slides contained the following table:

\(S\)	\(a(n)\)	Records
Fibonacci	A280521	A280523
\(n\)	A057945	A006893
\(\frac{n(n+1)}{2}\)	A281367	A281368
\(2^n\)	A100661	A000325
\(n^2\)	A280053	A280054

Related sequences

While writing this post, I submitted OEIS sequence A382814: the number of Nachos that the first player gets.

A good rule of thumb is that the player who draws last before the first reset gets the greatest number of nachos.

I have a conjecture that I bet you could prove: let \(n\) be the number of nachos in the starting pile. When \(n > 32\), the last player to take nachos before “the number of nachos left is less than the next Fibonacci number, [and] they start the sequence over” is the player that ends up with the greatest number of nachos in the end.

Prove it, contribute the proof to the OEIS, and let me know on Bluesky @peterkagey.com.