Timeline for The smallest sequence without differences among Fibonacci numbers
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 5, 2023 at 8:27 | comment | added | ho boon suan | Oh, it is on the OEIS if you remove the first 0 in the sequence: oeis.org/A001581 | |
| Oct 4, 2023 at 11:19 | comment | added | Roland Bacher | @hoboonsuan Thanks for these informations. By the way, density at most 1/5 is easy: Two consecutive terms of the sequence are at least at distance 4, never at distance 5 and since 4+4=8 is a Fibonacci number, three consecutive elements have diameter at least 10. This yields a density of at most 1/5 (in fact, one can easily lower the bound somewhat). | |
| Oct 4, 2023 at 8:18 | comment | added | ho boon suan | Ah, okay; this problem has been studied before under the name Fibonacci nim (confusingly, since there are two nim variations with this name, and we are interested in the less popular one). See Jeremy C. Pond and Donald F. Howells, “More on Fibonacci Nim”, Fibonacci Quarterly 1 (3) (1963), 61–62, where it is proven that the density of $A$ is at most $1/5$. Unfortunately, I could not find any literature citing this work or investigating this topic further. | |
| Oct 3, 2023 at 22:38 | comment | added | ho boon suan | Also, this problem has been studied when $S$ is the set of squares — Furstenberg and Sárközy proved that square-difference-free sets have natural density equal to zero. The game version of this is then the subtract-a-square game, and the greedy set corresponding to this game is given by OEIS A030193, which begins 0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, 44, 52, … . | |
| Oct 3, 2023 at 22:29 | comment | added | ho boon suan | This can be thought of in terms of a two-player game called a subtraction game, where given a set $S\subset\mathbf N$ called the subtraction set, two players have a heap of $n$ tokens between them, and are able to remove $s$ tokens during their turn for some $s\in S$; the last player to move is then the winner. The set $A$ in question is then the set of cold positions of this game when $S=\mathcal F$; that is, the set of numbers $n$ such that first player in a game of $n$ tokens can never win under optimal play. | |
| Oct 3, 2023 at 19:08 | comment | added | Roland Bacher | @FabiusWiesner The sequence contains odd terms (169 is the first odd term). It is therefore different. | |
| Oct 3, 2023 at 16:29 | comment | added | Fabius Wiesner | Starting from the second term $4$ and up to $128$ the sequence coincides with $2$ times OEIS A182771 i.e. $2\lfloor n(6+\sqrt{3})/3 \rfloor$. | |
| Oct 3, 2023 at 16:15 | comment | added | Roland Bacher | Nice observation! Thanks. | |
| Oct 3, 2023 at 15:58 | comment | added | Peter Taylor | Intuitively, it seems that the rarity of odd elements can be explained by the fact that two thirds of Fibonacci numbers are odd. This gives a bias towards even numbers at the start, and that bias then self-reinforces. | |
| Oct 3, 2023 at 15:15 | history | edited | Christian Remling | CC BY-SA 4.0 |
added 1 character in body
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| Oct 3, 2023 at 14:44 | history | asked | Roland Bacher | CC BY-SA 4.0 |