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.

It takes its name after its inventor , a Colombian mathematician.

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Definition

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

: $a_n = \begin 0 && \text n = 0 \\ a_ -n && \text a_ -n > 0 \text \\ a_ + n && \text \end$

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 .

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 title
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:

: $a_n \geq 0$
: $, a_n - a_, = n$
This is not a permutation of the integers: the first repeated term is $42 = a_ = a_$. Another one is $43 = a_ = a_$.

Conjecture

Neil Sloane has conjectured that every number eventually appears, 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.

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:

: $a_1 = 1$
: $a_ = \begin a_n / n && \text n \text a_n \\ n a_n && \text \end$

This OEIS entry is .

References

External links

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* {{youTube , FGC5TdIiT9U , The Slightly Spooky Recamán Sequence. (June 14, 2018) Numberphile

The Recamán's sequence
at Rosetta Code

The Ultimate Guide to Recamán’s Sequence (visualization, sonification, and animation)

Sequences and series
Integer sequences
Recurrence relations