### 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

Free Downloads:

STEM artwork

for your holiday gifts

Latest Books by Dev Gualtieri

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

Other Books

- Ultra-pure Green Light - November 20, 2017

- High Energy Cosmic Rays - November 13, 2017

- Advanced Aluminum Alloys - November 6, 2017

- Joseph Polchinski - October 30, 2017

- Our Magnetic Universe - October 23, 2017

- Cavitation - October 16, 2017

- Pell Numbers - October 9, 2017

- Miniature Antennas - October 2, 2017

- Fizzy Graphene - September 25, 2017

- The First Angiosperm - September 18, 2017

- Noise Thermometry and the Boltzmann Constant - September 11, 2017

- Walking in the Rain - September 4, 2017

- Agitated Atoms - August 28, 2017

- Partial Solar Eclipse at New Jersey - August 24, 2017

- Magnetocapacitive Tunnel Junctions - August 21, 2017

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**