In mathematics and computer science, Recamán's sequence is a sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.
Recamán's sequence was named after its inventor, Colombian mathematician , by Neil Sloane, creator of the On-Line Encyclopedia of Integer Sequences (OEIS). __TOC__
Definition Recamán's sequence is defined as:
The first terms of the 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, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155, ...
Visual representation
The most-common visualization of the Recamán's sequence is simply plotting its values, such as the figure seen here.
On January 14, 2018, the Numberphile YouTube channel published a video titled "The Slightly Spooky Recamán Sequence", showing a visualization using alternating semi-circles, as it is shown in the figure at the top of this page.
Sound representation
Values of the sequence can be associated with musical notes, in such that the running of the sequence can be associated with an execution of a musical tune.
Properties The sequence satisfies:
While the sequence attempts to avoid repeated terms while subtracting, it does not do so while adding, and repeats do in fact occur. Since there are repeated terms, this is not a permutation of the integers.
The first repeated term is . Another one is .
Conjecture Neil Sloane has conjectured that every number eventually appears, but this has not been proven. As of 2026, terms have been calculated, and 852,655 is the smallest natural number to not appear on the list.
References
External links
The Recamán's sequence at Rosetta Code The Ultimate Guide to Recamán’s Sequence (visualization, sonification, and animation)