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

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

A plot for the first 100 terms of the Recamán's sequence.^[4]

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

"Recamán's sequence"

"Play Recamán's sequence."

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]

a_{n}\geq 0

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

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

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:

a_{1}=1

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

^ ^a ^b ^c "A005132 - Oeis".
^ "Recamán's Sequence".
^ ^a ^b The Slightly Spooky Recamán Sequence, Numberphile video.
^ R.Ugalde, Laurence. "Recamán sequence in Fōrmulæ programming language". Fōrmulæ. Retrieved July 26, 2021.
^ "The On-Line Encyclopedia of Integer Sequences® (OEIS®)".
^ Math less traveled
^ "A057167 - Oeis".
^ "A064227 - Oeis".
^ "A064228 - Oeis".
^ 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

External links

Read other articles:

Filarkhos (gelar)

Untuk sejarawan Yunani Kuno, lihat Filarkhos (sejarawan). Filarkhos (bahasa Yunani Kuno: φύλαρχος, translit: phýlarkhos, Latin: phylarchuscode: la is deprecated ) adalah gelar yang dipakai pada zaman Yunani Kuno, berarti kepala suku, dari gabungan kata file, berarti suku + archein berarti memerintah. Di Athena klasik, seorang filarkhos adalah komandan kavaleri terpilih yang disediakan oleh masing-masing dari. sepuluh suku kota. Pada akhir periode Romawi abad ke-4 hingga ke-7 Mase...

Digamma function

Mathematical function For Barnes' gamma function of two variables, see double gamma function. The digamma function ψ ( z ) {\displaystyle \psi (z)} ,visualized using domain coloring Real part plots of the digamma and the next three polygamma functions along the real line In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:[1][2][3] ψ ( z ) = d d z ln ⁡ Γ ( z ) = Γ ′ ( z ) Γ...

Gianfelice Facchetti

Gianfelice Facchetti Gianfelice Facchetti (Milano, 28 agosto 1974[1]) è un attore, scrittore e regista teatrale italiano. È figlio di Giacinto Facchetti, calciatore dell'Inter e della Nazionale italiana[2][3]. Indice 1 Biografia 2 Spettacoli teatrali 3 Filmografia 4 Libri 5 Note 6 Altri progetti 7 Collegamenti esterni Biografia Mentre consegue la laurea in scienze dell'educazione, si forma presso la scuola di teatro di Quelli di Grock che frequenta per tre anni e con...

Orani João Tempesta

Orani João TempestaO.Cist.Kardinal, Uskup Agung São Sebastião do Rio de JaneiroGerejaGereja Katolik RomaTakhtaSão Sebastião do Rio de JaneiroPenunjukan27 Februari 2009Awal masa jabatan19 April 2009PendahuluEusébio Oscar Scheid, S.C.J.Jabatan lainKardinal-Imam Santa Maria Madre della Providenza a Monte Verde (2014–sekarang) Uskup Agung Belém do Pará (2004–2009) Administrator Apostolik Kawasan Biara Claraval (1999–2002) Uskup Rio Preto (1997–2004) Abbas Biara São Bernardo (1996...

Primera División 2009-2010 (Spagna)

Primera División 2009-2010Liga BBVA 2009-2010 Competizione Primera División Sport Calcio Edizione 79ª Organizzatore RFEF Date dal 29 agosto 2009al 16 maggio 2010 Luogo Spagna Partecipanti 20 Formula Girone all'italiana Risultati Vincitore Barcellona(20º titolo) Retrocessioni Real ValladolidTenerifeXerez Statistiche Miglior giocatore Lionel Messi[1] Miglior marcatore Lionel Messi (34) Miglior portiere Víctor Valdés[1] Incontri disputati ...

Ein bißchen Frieden

Ein bißchen Frieden Chanson de Nicole auConcours Eurovision de la chanson 1982 Sortie 1982 Auteur Bernd Meinunger Compositeur Ralph Siegel Chansons représentant l'Allemagne au Concours Eurovision de la chanson Johnny Blue(1981) Rücksicht(1983) Chansons ayant remporté le Concours Eurovision de la chanson Making Your Mind Up(1981) Si la vie est cadeau(1983)modifier Ein bißchen Frieden (« Un peu de paix ») est la chanson gagnante du Concours Eurovision de la chanson 1982, ...

C.M. Newton

Questa voce sugli argomenti cestisti statunitensi e allenatori di pallacanestro statunitensi è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti dei progetti di riferimento 1, 2. C.M. Newton Nazionalità Stati Uniti Altezza 188 cm Peso 86 kg Pallacanestro Ruolo Guardia / alaAllenatore Termine carriera 1951 - giocatore1989 - allenatore Hall of fame Naismith Hall of Fame (2000) Carriera Giovanili Fort Lauderdale High School19...

Hydrostatic test

Non-destructive test of pressure vessels For the autopsy procedure, see lung float test. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Hydrostatic test – news · newspapers · books · scholar · JSTOR (September 2009) (Learn how and when to remove this message) Hydrostatic tester A hydrostatic test is a way i...

Metropolitan Transport Corporation (Chennai)

Public transport in Chennai Metropolitan Transport Corporation (Chennai) Ltd.Mānakara Pōkkuvarattu Kaḻakam (Ceṉṉai)Formerly Pallavan Transport Corporation Dr. Ambedkar Transport Corporation ParentTamil Nadu State Transport Corporation Ltd.HeadquartersPallavan House, Anna Salai, Chennai, Tamil NaduLocaleChennai, Tamil NaduService areaChennai Metropolitan AreaService typeOrdinary, Express, Deluxe, Air Conditioned Deluxe and Small BusDaily ridership2.946 million per dayOperatorGovernment...

Emanuel Swedenborg

Emanuel SwedenborgLahir29 Januari 1688 KJ (8 Februari 1688)Stockholm, SwediaMeninggal29 Maret 1772London, InggrisEraFilsafat abad ke-18KawasanFilsafat BaratAliranTeosofi, mistisismeInspirasi atas Swedenborgianisme Dipengaruhi Plato, Aristoteles, Plotinus, Augustinus, Descartes, Leibniz Memengaruhi Kant, William Blake, August Strindberg, Charles Baudelaire, Carl Jung, Ralph Waldo Emerson, George Inness Emanuel Swedenborg pengucapanⓘ (nama lahir: Emanuel Swedberg; 8 Februari 1688 –...

Nicholas Georgescu-Roegen

Romanian mathematician, statistician and economist (1906–1994) The neutrality of this article is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until conditions to do so are met. (July 2022) (Learn how and when to remove this message) Nicholas Georgescu-RoegenNicolae GeorgescuBorn(1906-02-04)February 4, 1906Constanța, RomaniaDiedOctober 30, 1994(1994-10-30) (aged 88)Nashville, Tennessee, United StatesResting placeBellu Cemetery, Buchares...

Presidential Succession Act

Law regarding the order of succession to United States president The United States Presidential Succession Act is a federal statute establishing the presidential line of succession.[1] Article II, Section 1, Clause 6 of the United States Constitution authorizes Congress to enact such a statute: Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Off...

Masa Darurat (India)

Perdana Menteri Indira Gandhi, yang meminta Presiden India Fakhruddin Ali Ahmed memproklamasikan keadaan darurat nasional dari 25 Juni 1975 sampai 21 Maret 1977 Di India, Masa Darurat merujuk kepada periode 21 bulan dari 1975 sampai 1977 saat Perdana Menteri Indira Gandhi mendeklarasikan keadaan darurat di seluruh negeri. Resmi dikeluarkan oleh Presiden Fakhruddin Ali Ahmed di bawah Artikel 352(1) dari Konstitusi karena gangguan dalam negeri, Masa Darurat tersebut berlaku dari 25 Juni 1975 sa...

Johannes Jørgensen

Danish writer This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Johannes Jørgensen – news · newspapers · books · scholar · JSTOR (September 2019) (Learn how and when to remove this message) Jens Johannes JørgensenJørgensen about 1925Born(1866-11-06)6 November 1866Svendborg, DenmarkDied29 May 1956(1956-05-29...

United States at the 1948 Summer Olympics

Sporting event delegationUnited States at the1948 Summer OlympicsFlag of the United StatesIOC codeUSANOCUnited States Olympic Committeein LondonCompetitors300 (262 men and 38 women) in 19 sportsFlag bearerRalph CraigMedalsRanked 1st Gold 38 Silver 27 Bronze 19 Total 84 Summer Olympics appearances (overview)189619001904190819121920192419281932193619481952195619601964196819721976198019841988199219962000200420082012201620202024Other related appearances1906 Intercalated Games The United Sta...

Department of Homeland Security Office of Inspector General

Accountability and oversight bodyU.S. Department of Homeland Security Office of Inspector GeneralAgency overviewFormed2002TypeInspector General OfficeEmployees600+Annual budget$214.9m USD (FY 2023)Agency executivesJoseph Cuffari, Inspector GeneralGlenn Sklar, Principal Deputy Inspector GeneralGladys Ayala, Deputy Inspector General, IntegritySarah Nelson, Deputy Inspector General, InnovationBruce Miller, Deputy Inspector General, AuditsKristen Fredricks, Deputy Inspector General, External Affa...

MacOS

Operating system for Apple computers OSX and OS X redirect here. For other uses, see OSX (disambiguation). This article is about macOS version 10.0 and later. For Mac OS 9 and earlier, see Classic Mac OS. For the family of Mac operating systems, see Mac operating systems. For the Ugandan school nicknamed Macos, see Makerere College School. Operating system macOSmacOS Sonoma, the latest release of macOSDeveloperAppleWritten in C C++[1] Objective-C Swift[2] assembly language OS ...

Rumah Pilkada

Rumah PilkadaPembuatRosianna SilalahiSri SumarniPengembangRosianna SilalahiSri SumarniPresenterAdisty Larasati & Pascalis Iswari (2018)Audrey Chandra & Muhammad Syahreza (2020)Negara asalIndonesiaBahasa asliBahasa IndonesiaProduksiDurasi30 menit (Senin-Jumat)Rilis asliJaringanKompas TVFormat gambar1080p HDTVFormat audioStereoRilis08 Januari 2018 –sekarangAcara terkaitKompas, Rumah Pemilu Rumah Pilkada adalah salah satu acara televisi dari stasiun televisi Kompas TV. Acara ini ...

هامبورغ (آيوا)

لمعانٍ أخرى، طالع هامبورغ (توضيح). هامبورغ الإحداثيات 40°36′21″N 95°39′18″W / 40.605833333333°N 95.655°W / 40.605833333333; -95.655 [1] تقسيم إداري البلد الولايات المتحدة[2] التقسيم الأعلى مقاطعة فريمونت خصائص جغرافية المساحة 2.84509 كيلومتر مربع2.845088 ك�...

Charles Dickens

Vous lisez un « article de qualité » labellisé en 2013. Il fait partie d'un « thème de qualité ». Pour les articles homonymes, voir Dickens. Charles Dickens Charles Dickens à New York en 1868. Données clés Nom de naissance Charles John Huffam Dickens Naissance 7 février 1812 Landport, près de Portsmouth, Royaume-Uni Décès 9 juin 1870 (à 58 ans) Gad's Hill Place à Higham (Kent) au Royaume-Uni Nationalité Britannique Activité principale Romanci...