A382814 Number of nachos that the first player gets when playing the "Fibonachos" game starting with n nachos.

1, 1, 2, 3, 3, 4, 3, 4, 4, 5, 6, 8, 8, 9, 9, 9, 10, 10, 12, 8, 9, 9, 10, 11, 11, 12, 11, 12, 12, 13, 14, 16, 21, 21, 22, 22, 22, 23, 23, 25, 25, 26, 26, 26, 25, 26, 26, 27, 28, 28, 29, 28, 33, 21, 22, 22, 23, 24, 24, 25, 24, 25, 25, 26, 27, 29, 29, 30, 30, 30

Offset: 1

Author

Peter Kagey, Apr 05 2025

Keywords

• nonn

Comments

From the Fibonachos link: "Two people are sharing a plate of nachos. They take turns dividing the nachos, each taking the n-th Fibonacci number of nachos on the n-th 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?"

The first ten terms of the sequence are the following:

a(1) = 1 via [[1]];

a(2) = 1 via [[1, 1]];

a(3) = 1 + 1 = 2 via [[1, 1], [1]];

a(4) = 1 + 2 = 3 via [[1, 1, 2]];

a(5) = 1 + 2 = 3 via [[1, 1, 2], [1]];

a(6) = 1 + 2 + 1 = 4 via [[1, 1, 2], [1, 1]];

a(7) = 1 + 2 = 3 via [[1, 1, 2, 3]];

a(8) = 1 + 2 + 1 = 4 via [[1, 1, 2, 3], [1]];

a(9) = 1 + 2 + 1 = 4 via [[1, 1, 2, 3], [1, 1]]; and

a(10) = 1 + 2 + 1 + 1 = 5 via [[1, 1, 2, 3], [1, 1], [1]].

Conjecture:

a(n) = n/2 if and only if n is in {2, 8, 10, 32}.

Conjecture:

For n > 32, a(n) > n/2 if and only if F(m)-1 <= n <= F(m+1)-2 for some odd integer m, where F(n) = A000045(n).

Links

• Neil J.A. Sloane, Winter Fruits: New Problems from OEIS. (Sequence mentioned from 12:50-19:50.)
• Reddit user Teblefer, Fibonachos

Crossrefs

Cf. A000045, A280521.

Programs

Mathematica

A382814[n_] := Module[{m = n, i = 1, p = True, c = 0},
   While[m > 0,
      If[Fibonacci[i] > m, i = 1];
      If[p, c += Fibonacci[i]];
      m -= Fibonacci[i]; i += 1; p = Not[p];
   ];
   c
];