Recamán's sequence

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 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 … {\displaystyle a_{0},a_{1},a_{2}\dots } is defined as:

a n = { 0 if n = 0 a n − 1 − n if a n − 1 − n > 0 and is not already in the sequence a n − 1 + n otherwise {\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)

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:

a n ≥ 0 {\displaystyle a_{n}\geq 0}

| a n − a n − 1 | = n {\displaystyle |a_{n}-a_{n-1}|=n}

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

Conjecture Neil Sloane has conjectured that every number eventually appears, 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.

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

References

External links 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)