Recamán's sequence

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.

A drawing of the first 75 terms of Recamán's sequence, according to the method of visualization shown in the Numberphile video "The Slightly Spooky Recamán Sequence"

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).

Definition

Recamán's sequence $a_0,a_1,a_2\dots$ is defined as:

$a_n=\{\begin{array}{ccc}0& & \text{if }n=0\\ a_{n-1}-n& & \text{if }{a}_{n-1}-n>0\text{ and is not already in the sequence}\\ a_{n-1}+n& & \text{otherwise}\end{array}$

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)

Visual representation

A plot for the first 100 terms of the Recamán's sequence.

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.

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.

Properties

The sequence satisfies:

$a_n\ge 0$

$|a_n-a_{n-1}|=n$

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

Conjecture

Neil Sloane has conjectured that every number eventually appears, but this has not been proven. As of 2018, 10²³⁰ terms have been calculated, and 852,655 is the smallest natural number to not appear on the list.

Uses

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

Alternate sequence

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

$a_1=1$

$a_{n+1}=\{\begin{array}{ccc}{a}_n/n& & \text{if }n\text{ divides }{a}_n\\ n{a}_n& & \text{otherwise}\end{array}$

This OEIS entry is A008336.

External links

• OEIS sequence A005132 (Recamán's sequence)
• 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)