### 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.

*Permanent Link to this article*

Linked Keywords: STEM fields; mathematics; number theory; parent; child; children; natural number; Unicode; integer sequence; A000027; The On-Line Encyclopedia of Integer Sequences; computer scientist; A001477; collegiality; colleague; A181975; daydream; sleep; A001745; boredom; boring; prime number; A000040; divisor; arXiv; mathematician; Neil J. A. Sloane; A098550; Cartesian coordinate system; graph; geyser; Yellowstone National Park; park; U.S. state; Wyoming; parity; even; odd; Old Faithful geyser; meter; Wikimedia Commons; greatest common denominator; chart pattern; price chart; volatility; volatile; stock; Gnumeric; computer; experimental mathematics; computer programming; C language; a098550.c; caveat; professional; programmer; desktop computer; David Applegate; C++; mathematical jargon; proof by inspection; straight line; slope; hypothesis; analysis; A005132.

### Google Search

Latest Books by Dev Gualtieri

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- Larry Tesler (1945-2020) - March 30, 2020

- Power from Raindrops - March 23, 2020

- A Voice from the Crypt - March 19, 2020

- Computer Hashing - March 9, 2020

- Carbon Capture - March 2, 2020

- Knots - February 24, 2020

- When in Rome... - February 17, 2020

- Frigid Legos® - February 10, 2020

- Natural Materials - February 3, 2020

- Martian Atmosphere - January 27, 2020

- Whiskey - January 20, 2020

- An Elastocaloric Calorimeter - January 13, 2020

- Elastocaloric Effect - January 6, 2020

- Christmas 2019 - December 23, 2019

- Our Accelerating Universe - December 16, 2019

- Math is Beautiful - December 9, 2019

- Giovanni Battista Riccioli - December 2, 2019

- Consider a Spherical Turkey - November 25, 2019

- Metamaterial Lenses - November 18, 2019

- Computing Chaos - November 11, 2019

- Ultra-Hyper-Super Black - November 4, 2019

- Interstellar Comets - October 28, 2019

- Magnetic Pole Flip - October 21, 2019

- Calvin Quate (1923-2019) - October 14, 2019

- Transistor Threads - October 7, 2019

- Scientific Broadness - September 30, 2019

- Squaring the Circle - September 23, 2019

- Death by Air Pollution - September 16, 2019

- Lots, and Lots, of Dots - September 9, 2019

- Fiber-Optic Sensors - September 2, 2019

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**