Stern Brocot Series
The Stern-Brocot sequence (A002487 in OEIS) also known as the "diatomic sequence of Stern and Brocot" or "star sequence" is a sequence of whole numbers that was independent of the mathematician Moritz Stern and the watchmaker Achille Brocot (also known by the Brocot inhibition ). Among other things, it is the basis of the Stern-Brocot tree for counting rational numbers.
Brocot, as a watchmaker, came across the result when he was looking for the most suitable gear pairing (more precisely: the best approximation by a ratio of the whole number of teeth) for a desired gear ratio (decimal).
The education law of the following is:
The sequence of numbers is 0; 1; 1; 2; 1; 3; 2; 3; 1; 4; 3; 5; 2; 5; 3; 4; 1; 5; 4; 7; ... whereby the following step table (beginning of the sequence with s 1 ) is used to better represent various properties:
| 1 | |||||||||||||||||||
| 1 | 2 | ||||||||||||||||||
| 1 | 3 | 2 | 3 | ||||||||||||||||
| 1 | 4th | 3 | 5 | 2 | 5 | 3 | 4th | ||||||||||||
| 1 | 5 | 4th | 7th | 3 | 8th | 5 | 7th | 2 | 7th | 5 | 8th | 3 | 7th | 4th | 5 | ||||
| 1 | 6th | 5 | 9 | 4th | 11 | 7th | 10 | 3 | 11 | 8th | 13 | 5 | 12 | 7th | 9 | 2 | 9 | 7th | ... |
| 1 | 7th | 6th | 11 | 5 | 14th | 9 | 13 | 4th | 15th | 11 | 18th | 7th | 17th | 10 | 13 | 3 | 14th | 11 | ... |
| 1 | 8th | 7th | 13 | 6th | 17th | 11 | 16 | 5 | 19th | 14th | 23 | 9 | 22nd | 13 | 17th | 4th | 19th | 15th | ... |
| 1 | 9 | 8th | 15th | 7th | 20th | 13 | 19th | 6th | 23 | 17th | 28 | 11 | 27 | 16 | 21st | 5 | 24 | 19th | ... |
The sum of the 2 n numbers in the nth row is 3 n, and if n is a power of two, then s n = 1. Each of the columns is an arithmetic sequence ; H. the difference between two numbers below each other is constant.
Connection with other consequences
If the Pascal triangle is arranged as a table of levels (see following table), the ascending diagonals have two properties: The number of odd numbers is the Stern-Brocot sequence, the sum of the numbers is the Fibonacci sequence .
| 1 | ||||||
| 1 | 1 | |||||
| 1 | 2 | 1 | ||||
| 1 | 3 | 3 | 1 | |||
| 1 | 4th | 6th | 4th | 1 | ||
|---|---|---|---|---|---|---|
| 1 | 5 | 10 | 10 | 5 | 1 | |
| 1 | 6th | 15th | 20th | 15th | 6th | ... |
| 1 | 7th | 21st | 35 | 35 | 21st | ... |
| 1 | 8th | 28 | 56 | 70 | 56 | ... |
| 1 | 9 | 36 | 84 | 126 | 126 | ... |
The diagonal marked yellow shows an example. In the first column one starts from row n = 9 and considers the other values of the diagonals: The number of odd numbers is 4 and s 9 is also 4; The sum on the diagonal is 1 + 7 + 15 + 10 + 1 = 34 = f 9 in the Fibonacci sequence.
Web links
- Code examples for programming the Stern-Brocot sequence on Rosettacode.org
- Article “ The misunderstood sister of the Fibonacci sequence ” by Jean-Paul Delahaye in “Spectrum of Science” 05/2015 (only as an excerpt, but with the sources of the article and further links).
Individual evidence
- ↑ A002487 is the name of the sequence of numbers in the On-Line Encyclopedia of Integer Sequences ( http://oeis.org )
- ↑ Article “About a number-theoretic function.” By Moritz Stern, published in “Journal for pure and applied mathematics”, 55, 1858. pp. 193–220.
- ↑ Article “Calcul des rouages par approximation, Nouvelle méthode.” By Achille Brocot, published in “Revue chronomeétrique”, 3, 1861. 186-94
- ↑ The application is described, for example, in the article " Trees, Teeth, and Time: The mathematics of clock making " by David Austin, published by the American Mathematical Society (accessed May 15, 2015)