Recamán's sequence 📖 Wikipedia

In mathematics and computer science, Recamán's sequence^[1][2] 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 Bernardo Recamán Santos [es], by Neil Sloane, creator of the On-Line Encyclopedia of Integer Sequences (OEIS).

Recamán's sequence ${\displaystyle a_{0},a_{1},a_{2}\dots }$ is defined as:

${\displaystyle a_{n}={\begin{cases}0&&{\text{if }}n=0\\a_{n-1}-n&&{\text{if }}a_{n-1}-n>0{\text{ and is not already in the sequence}}\\a_{n-1}+n&&{\text{otherwise}}\end{cases}}}$

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, ... (sequence A005132 in the OEIS)

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",^[3] showing a visualization using alternating semi-circles, as it is shown in the figure at top of this page.

"Recamán's sequence"

"Play Recamán's sequence."

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.^[5]

The sequence satisfies:^[1]

${\displaystyle a_{n}\geq 0}$

${\displaystyle |a_{n}-a_{n-1}|=n}$

This is not a permutation of the integers: the first repeated term is ${\displaystyle 42=a_{24}=a_{20}}$.^[6] Another one is ${\displaystyle 43=a_{18}=a_{26}}$.

Neil Sloane has conjectured that every number eventually appears,^[1] but this has not been proven. As of 2026, 10⁶¹² terms have been calculated, and 852,655 is the smallest natural number to not appear on the list.^[1]

Besides its mathematical and aesthetic properties, Recamán's sequence can be used to secure 2D images by steganography.^[7]

• OEIS sequence A005132 (Recamán's sequence)
• Weisstein, Eric W. "Recamán's Sequence". MathWorld.
• The Slightly Spooky Recamán Sequence. (June 14, 2018) Numberphile on YouTube
• The Recamán's sequence at Rosetta Code
• The Ultimate Guide to Recamán’s Sequence (visualization, sonification, and animation)