# Recamán's sequence

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.

It takes its name after its inventor Bernardo Recamán Santos [es], a Colombian mathematician.

## Definition

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, ...

## On-line encyclopedia of integer sequences (OEIS)

Recamán's sequence was named after its inventor, Colombian mathematician Bernardo Recamán Santos, by Neil Sloane, creator of the On-Line Encyclopedia of Integer Sequences (OEIS). The OEIS entry for this sequence is A005132.

## Visual representation

The most-common visualization of the Recamán's sequence is simply plotting its values, such as the figure at right.

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.

## Sound representation

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.[5]

## Properties

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 ${\displaystyle 43=a_{18}=a_{26}}$.

### Conjecture

Neil Sloane has conjectured that every number eventually appears,[7][8][9] but it has not been proved. Even though 10230 terms have been calculated (in 2018), the number 852,655 has not appeared on the list.[1]

### Uses

Besides its mathematical and aesthetic properties, Recamán's sequence can be used to secure 2D images by steganography.[10]

## Alternate sequence

The sequence is the most-known sequence invented by Recamán. There is another sequence, less known, defined as:

${\displaystyle a_{1}=1}$
${\displaystyle a_{n+1}={\begin{cases}a_{n}/n&&{\text{if }}n{\text{ divides }}a_{n}\\na_{n}&&{\text{otherwise}}\end{cases}}}$

This OEIS entry is A008336.

## References

1. ^ a b c "A005132 - Oeis".
2. ^
3. ^ a b The Slightly Spooky Recamán Sequence, Numberphile video.
4. ^ R.Ugalde, Laurence. "Recamán sequence in Fōrmulæ programming language". Fōrmulæ. Retrieved July 26, 2021.
5. ^
6. ^ Math less traveled
7. ^
8. ^
9. ^
10. ^ S. Farrag and W. Alexan, "Secure 2D Image Steganography Using Recamán's Sequence," 2019 International Conference on Advanced Communication Technologies and Networking (CommNet), Rabat, Morocco, 2019, pp. 1-6. doi: 10.1109/COMMNET.2019.8742368