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

%I A382814 #10 Apr 12 2025 12:38:46
%S A382814 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,
%T A382814 13,14,16,21,21,22,22,22,23,23,25,25,26,26,26,25,26,26,27,28,28,29,28,
%U A382814 33,21,22,22,23,24,24,25,24,25,25,26,27,29,29,30,30,30
%N A382814 Number of nachos that the first player gets when playing the "Fibonachos" game starting with n nachos.
%C A382814 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?"
%C A382814 The first ten terms of the sequence are the following:
%C A382814 a(1) = 1                  via [[1]];
%C A382814 a(2) = 1                  via [[1, 1]];
%C A382814 a(3) = 1 + 1 = 2          via [[1, 1], [1]];
%C A382814 a(4) = 1 + 2 = 3          via [[1, 1, 2]];
%C A382814 a(5) = 1 + 2 = 3          via [[1, 1, 2], [1]];
%C A382814 a(6) = 1 + 2 + 1 = 4      via [[1, 1, 2], [1, 1]];
%C A382814 a(7) = 1 + 2 = 3          via [[1, 1, 2, 3]];
%C A382814 a(8) = 1 + 2 + 1 = 4      via [[1, 1, 2, 3], [1]];
%C A382814 a(9) = 1 + 2 + 1 = 4      via [[1, 1, 2, 3], [1, 1]]; and
%C A382814 a(10) = 1 + 2 + 1 + 1 = 5 via [[1, 1, 2, 3], [1, 1], [1]].
%C A382814 Conjecture:
%C A382814 a(n) = n/2 if and only if n is in {2, 8, 10, 32}.
%C A382814 Conjecture:
%C A382814 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).
%H A382814 Neil J.A. Sloane, <a href="https://vimeo.com/201218446">Winter Fruits: New Problems from OEIS</a>. (Sequence mentioned from 12:50-19:50.)
%H A382814 Reddit user Teblefer, <a href="https://www.reddit.com/r/math/comments/5lxh3c">Fibonachos</a>
%t A382814 A382814[n_] := Module[{m = n, i = 1, p = True, c = 0},
%t A382814    While[m > 0,
%t A382814       If[Fibonacci[i] > m, i = 1];
%t A382814       If[p, c += Fibonacci[i]];
%t A382814       m -= Fibonacci[i]; i += 1; p = Not[p];
%t A382814    ];
%t A382814    c
%t A382814 ];
%Y A382814 Cf. A000045, A280521.
%K A382814 nonn
%O A382814 1,3
%A A382814 _Peter Kagey_, Apr 05 2025