### The Yellowstone Sequence

February 12, 2015 People in STEM fields learn at lot of mathematics in their lifetime,but it all starts with number theory. Parents teach their two-year-old children, "one-two-three...," along with "a,b,c..." The natural numbers, as symbolized by Unicode character, U+2115, shown at the left, are assigned integer sequence A000027 in The On-Line Encyclopedia of Integer Sequences. |

*Old Faithful geyser in Yellowstone National Park, WyomingOld Faithful, as its name implies, is one of the more predictable geysers. It erupts to a height of 32-56 meters for 1.5-5 minutes in intervals of about 45 to 125 minutes.(Photo by Kamilo Kardona, via Wikimedia Commons.)*

The construction of integer sequence A098550 is as follows. Sequence elements a(n) are equal to n if n is less than or equal to 3. Otherwise, the next sequence element a(n) is the smallest number not present in the sequence up to that point having at least one common factor with a(n-2), but none with a(n-1). If we have a function, GCD() that returns the greatest common denominator of its two arguments, then the sequence element a(n) would have GCD(a(n), a(n-1)) = 1 and GCD(a(n), a(n-2)) > 1. The first few elements of integer sequence A098550 are 1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17, 18, and 85, and a graph of its first hundred elements appears below.

*Graph of the first hundred terms of integer sequence A098550.It looks to me more like the price chart for a volatile stock than geyser eruptions.(Plot rendered by the author using Gnumeric.)*

computers have enabled the practice of experimental mathematics, and we can better see some properties of this sequence if we crank out more terms. For this purpose, I've written a program in the C language (source code, here). My usual caveats, that I'm not a professional programmer and that there are likely better ways to write such a program, apply. However, the program is short, easily understood, and it computed 20,000 elements in just a few seconds on my desktop computer. David Applegate, another author of the arXiv paper, has written a more elegant C++ program.[2] The first 10,000 terms of integer sequence A098550, as calculated by my program, appear in the plot below.

(Plot rendered by the author using *Gnumeric.)(Click for larger image)*

As can be seen by inspection, the elements of the sequence seem to fall on straight lines. When a quarter of a million points are analyzed, the slopes of these lines are approximately 0.467, 0.957, 1.15, 1.43, 2.40, 3.38, 5.25 and 6.20. You can just barely see traces of the eighth line in the above graph. Most interestingly, if we ignore the line with slope 1.15, the ratios of the other slopes are quite nearly consecutive primes.

Line | Slope | Slope(n)/Slope(n-1) | |

A | 0.467 | ||

B | 0.957 | 2.049 | |

D | 1.430 | 3.062 | |

E | 2.400 | 5.139 | |

F | 3.380 | 7.238 | |

G | 5.250 | 11.242 | |

H | 6.200 | 13.276 |

a(0) = 0The first few terms of this sequence are 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, and 77. Unlike the Yellowstone sequence, which has each natural number included just once, this sequence has duplicates. The first example of this is that terms a(20) and a(24) are both 42. A graph of the first 100,000 elements appears below.

For n > 0:

a(n) = a(n-1) - n if positive and not already in the sequence

Otherwise a(n) = a(n-1) + n.

*Graph of 100,000 terms of integer sequence A005132 (Recamán's sequence).(Plot rendered by the author using Gnumeric.)*

### References:

- David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, "The Yellowstone Permutation," arXiv, January 7, 2015.
- David Applegate, A C++ program to compute terms of integer sequence A098550.
- N. J. A. Sloane, The first 100000 terms of integer sequence A005132.

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