In mathematics and computer science, Recamán's sequence is a well known 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__
Recamán's sequence
a0,a1,a2...
an=\begin{cases} 0&&ifn=0\\ an-n&&ifan-n>0andisnotalreadyinthesequence\\ an+n&&otherwise \end{cases}
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, ...
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 top of this page.
Values of the sequence can be associated with musical notes, in such that case the running of the sequence can be associated with an execution of a musical tune.[1]
The sequence satisfies:
an\geq0
|an-an-1|=n
42=a24=a20
43=a18=a26
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.
Besides its mathematical and aesthetic properties, Recamán's sequence can be used to secure 2D images by steganography.[3]