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color push Black color pop" N G cmbx12NumericalzAnaloguesofAronson'sSequence "@ cmti12Benoit35Cloitrffe ߚXQ cmr1213ruePinaigrier TVours37000,FRANCE (Email:8abmrt@wanadoSo.fr) N.35J.A.Sloffane BAVT&TShannonLabs FlorhamPrark,NJ07932{0971,USA F(Email:8njas@researcrh.att.com) Matthew35J.Vandermast Ϲ53PiagetAvrenue =Clifton,NJ07011{1216,USA m(Email:8ghoSdges14@msn.com)+ =-N cmbx12Abstract"j8Aronson'sksequence1,4,11,16,g cmmi12:::0iskdenedbrytheEnglishsentence\tistherst,fourth,elevrenth,sixteenth,:::kletterofthissenrtence."
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cmsy10 1)]whicrhisconsistentwiththecondition\nisamembSerofthesequence!ifandonlyifa(n)isoSdd."zThissequencecanalsobecrharacterizedbyits\square",8thefsequencea2|{Y cmr8(2)\|(n)Z=a(a(n)),whicrhfequals2n`E+3ffornZ1.YTherefaremanryF{generalizationsofthissequence,]psomeofwhicharenew,]pwhileothersthrownewlighrtonpreviouslyknownsequences._html: html:1.1Inutro =ductionb#Aronson'sqsequenceisdenedbrytheEnglishsentence\tistherst,cfourth,eleventh,sixteenrth,1:::letter#ofthissentence(notcountingspacesorcommas),"1andisaclassicexamplevXofaself-referenrtialsequence([html:color push cmyk 0 1 0 03 html: color pop],[html:color push cmyk 0 1 0 08 html: color pop],sequencevXM3406in[html:color push cmyk 0 1 0 013 html: color pop],fhtml:color push cmyk 0 1 0 0A5224 html: color popin[html:color push cmyk 0 1 0 012 html: color pop]).ItissomewhatunsatisfactorybSecauseoftheamrbiguityintheEnglishnamesfornrumbersorver100|forexample,%somepSeoplesary\onehundredandone",%whileotherssay\onehrundredone."IAnotherwell-knownexampleisGolomb'ssequence,2inwhichthen2thtermG(n)(fornUR1)isthenrumbSeroftimesnappearsinthesequence(fhtml:color push cmyk 0 1 0 0A1462 html: color popin[html:color push cmyk 0 1 0 012 html: color pop]):4i1;2;2;3;3;4;4;4;5;5;5;6;6;6;6;7;7;7;7;8;::: t*color push Black color pop *\͍>썍t*color push Blackhtml:color push gray 0 color pop html: color popn썑t*ThereisasimpleformrulaforG(n):8itisthenearestintegerto(andapproaches) Ӏ2K cmsy8 2 cmmi8jn 1T;t*whereUR=(1+plj z 95)=2([html:color push cmyk 0 1 0 05 html: color pop],[html:color push cmyk 0 1 0 06 html: color pop,SectionE25]). Additional7>examplescanbSefoundinHofstadter'sbooks[html:color push cmyk 0 1 0 07 html: color pop],Jd[html:color push cmyk 0 1 0 08 html: color pop]7>andin[html:color push cmyk 0 1 0 06 html: color pop]and[html:color push cmyk 0 1 0 012 html: color pop].t*Horwever,[the7sequencefa(n)gmenrtionedintheAbstractappSearstobenew,[asdomanryt*ofttheothersequenceswrewilldiscuss.$WVewillalsogivenewpropSertiesofsomesequencest*thatharvebSeenstudiedelsewhere. Sectionhtml:color push cmyk 0 1 0 02 html: color popdiscussesthesequencemenrtionedintheAbstract,handalsointroSducesthet*\square"ofasequence.NSomesimplegeneralizations(non-monotonic,\evren"and\lying"t*vrersions)ԥaredescribSedinSectionhtml:color push cmyk 0 1 0 03 html: color pop.4Theoriginalsequenceisbasedonexaminationofthet*sequencer+moSdulo2.InSectionhtml:color push cmyk 0 1 0 04 html: color popwreconsidervXarious\modyn9"generalizations.Sectionhtml:color push cmyk 0 1 0 05 html: color popt*extendsl&bSoththeoriginalsequenceandthe\modyn9"generalizationsbrydeningthet*\Aronsonrmtransform"ofasequence./FinallyV,^Sectionhtml:color push cmyk 0 1 0 06 html: color popbrie
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\Monotonicallyincreasing"meansthateacrhtermisstrictlygreaterthanthet*previousterm.8!
msbm10PUR=f1;2;3;:::ʞg,NUR=f0;1;2;3;:::ʞg.t*Vhtml: html: 2.1$g G cmmi12nzisinthesequenceifandonlyifaDt G G cmr17(n)iso =ddb#Letthesequencea(1),.a(2),a(3);:::sbSedenedbrytherulethata(n)isthesmallestpSositivreinteger>a(n 1)whicrhisconsistentwiththeconditionthat H html:H\nisamemrbSerofthesequenceifandonlyifa(n)isodd."w;, html:_color push Black(1) color popThe`
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bSe1,}csince1isoddand1wrouldbeinthesequence.It couldalsobSe2,]sincethen1wrouldnotbeinthesequence(becausethetermsmrustincrease)and2isevren.!Butwemusttakethesmal lestpSossiblevXalue,soa(1)UR=1.!Nowa(2)\cannotbSe2,*because2isevren.Norcana(2)be3,*forthen2wrouldnotbeinthesequencebuta(2)wrouldbSeodd.JHorwever,2
a(2)=4ispermissible,2
sowremusttakea(2)UR=4,andthen2and3arenotinthesequence.Soa(3)mrustbSeevenand>4,ɶanda(3)=6wrorks.P5Now4isinthesequence,ɶsoa(4)KmrustbSeodd,+anda(4)UR=7Kwrorks.Continuinginthiswaywendthattherstfew t*color push Black t2 color pop K\͍>썍t*color push Blackhtml:color push gray 0 color pop html: color popn썑t*termsareasfollorws(thisisfhtml:color push cmyk 0 1 0 0A79000 html: color pop):ʱdrvnUR: 91 52 13 -4 ɓ)5 s%6 378
9* 10@S11V12kc(Ia(n)UR: 91 54 16 -7 ɓ)8 s%9 S!11 1315* 16@S17V18k Oncewrearepasta(2)therearenofurthercomplications,Ya(n 1)isgreaterthann, t*andwrecffan,andthereforemust,takeʲ U html: