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Chapter 5 Numbers that count（第1页）

Chapter5t

&ariseoftheirownagproblemsareimportantaeigated。HereIwilldescribethebis，andthealan，FibonadStirliheyeainnaturals。Butwefirstbeginwithsomeveryfualnumbersequences。

Triangularhmetietricprogressions

&heyearwhebis，Iwilltakeamomehetriangularhofwhiotedbythesumofthefirstnumbers。Itsvalue，intermsofn，befoundbythefollowingtrick。Wewritetmehenagainasthesamesumbutinthereversethetwoversionsofther:

tn=1+2+3+···+（n-2）+（n-1）+n

tn=n+（n-1）+（n-2）+···+3+2+1;

Forexample，bytakinga=1athesumofthefirstnoddnumbersisn+n（n-1）=n+n2-hsquare。

Ifwereplaultipliastheoperation，wemovefromarithmeticseriestogeometricseries。Inaicseries，eachpairofsuccessivetermsisseparatedbyaohenumberbinournotation。Inotherwords，tomovefromohe，wesimplyaddb。Iricseries，weonbeginwithsomearbitraryhefirsttermandmovefromohebymultiplyingbyafixednumber，calledtheonratio，dehesymbolr。Thatistosay，thetypietricserieshastheforma，ar，ar2，···withthenthtermbeingarn-1。Aswitharithmeticseries，thereisaformulaforthesumofthefirstntermsofageometricseries:

Thequickwayofseeingthatthisformulaisrightistotaketheequivalentformthatweobtaiiplybothsidesofthisequationby（r-1）andmultiplyoutthebratheleft-haain:

（ar+ar2+ar3+···+arn）-（a+ar+ar2+···+arn-1）

andthewholeexpressionteleseaningthatermiscelledbyoherbracket:theoionsarearn-a=a（rn-1），showingthatourformulaforthesumiscorreple，puttinga=1ahesumofpowersof2:

1+2+4+···+2n-1=2n-1。

ThisformulaisjustwhatyouneedioverifyEuclid’sresultfromChapter3onhowtogenumbersfromMersenneprimes。

Factorials，permutations，andbis

Aswehaveseeriangularnumberarisesfromsummingallthenumbersfrom1uptoher。Ifwereplaultiplithisidea，wegetwhatareknoworialnumbers，whichmadetheirfirstappearaer2。

Faeuptlyingandprobabilityproblemssuchasthecesofbeiatypeofhandinacardgamelikepoker。Forthatreasoheirownnotatioorialisdenotedbyn!=n×（n-1）×···×2×1。Thetriangularnumbersgrowreasonablyquickly，atabouthalftherateofthesquares，butthefactrowmuchfasterandsoonpassintothemillionsandmillions:forexample10!=3，628，800。Theexarkalertsustothisratheralarmih。

&specialclassthatemergesingproblems，oreionsastheyarecalled，isthatofthebis，soheyariseasthemultipliersofpowersofxwhenthebinomialexpression（1+x）nisexpahebi，r）isthenumberofdifferentwayswemaystructasetofsizerfromoneofsizen。Forexample，C（4，2）=6，astherearesixpairs（takenwithardtoorderair）thatagroupoffour:forexample，ifwehavefour，Alex，Barbara，e，andDavid，therearesixwaysthatweselethisgroup:AB，AC，AD，Bdomialtsbecaltwodistinctways。First，wedtheargumeweusedtocalculatege，r）r!，whisgivesustheusefulexpression:

Thisfactorial-basedformulaforgbisdoesgiveanicealgebraithebisthatallowsustodemoheirmanyspecialproperties。However，theevolutioiesisofteraifwefoasedwaytogeegers，whichisbymeaigle（seeFigure2），alsoknoascal’sTriangle，inhonourofthe17th-turyFreidphilosopherBlaisePascal（1623–62）。（TheArithmetiglehasbeendisdre-dischoutPersia，India，andaoverthelast1，000years:forexample，itfeaturedasthefrohePreirrorbyChuShih-1303。）

2。TheArithmetigle

Eaumberinthebodyleisthesumofthetwoaboveit。Thetriangle，whibeuediely，givesthefulllistofbis。Forexample，tofindthenumberofwaysofselegfiveperoupofseven，proceedasfollows。heliriangle，beginning。Similarlyionswithinealefttainstartingwith0。Godowntothelinenumbered7，ahelinenumbered5（rememberingtostartyour0）:weseetheansweris21。Youwillryofeaple，21isalsothenumberofwaysofgtwoperoupofseven。Thisisexplainedbythatwhehefivefromseveaneouslygtwofromsevewobeingthepairleftbehind。Thissymmetryargumentofcourseappliestoeveryrow。Thisisalsomaheformulaonpage55，foritreturnsthesameexpressionifwerepla-r，asthetermsrandn-rthatweseeiorsimplysositions。

&hatthepatterherightahardtosee。Eachrowbuildsfromtheo。Weseeeasilythatthefirstthreerowsareple，the2ihethirdrowtellsusthattherearetwowaysofgasinglepersonfromapair。The1thatsitsontopissayingthatthereisoochooseasetofsizezerofromtheemptyset。Infact，thereisonewayofgasetofsizezerofroma，whichiswhyeveryrowbeginswith1。Letusfotheexamplejustgiven–thereare21=15+6waysofselegfivefromagroupofsevehe21quiurallysplitintotwotypes。First，thereare15waystroupoffourfromthefirstsixpeople，towhichwemayaddtheseveoformourfivesome。Ifwedon’tihpersoheobuildasetoffivefromthefirstsix，andtherearesixwaysofdoingthis。Thisillustrateshowoheryisthesumofthetwoaboveit，andthispatternpropagatesthroughle。Insymbolsthisruletakestheform:

-1，r）+-1，r-1）。

&riangleisripatterns。Forexample，summingallthenumbersineachsuccessiverowgivesthedoublingsequence1，2，4，8，16，32，···:thesequenceofpowersof2。Insummibegins1，8，28，56，···forinstance，wearesummingthenumberofwaysofgasetofsize0，1，2，3etasetof8。Intotal，thisgivesusthenumberofwaysofselegasetofanysizefromagroupof8，whichisequalto28as，ingeneral，asetofsizens2hinit。

Thislastfabeseely，forasubsetofasetofsizeifiedbyabinarystrihninthefolloesiderthesetiioninaspecificorder{a1，a2，···，an}say，andthenabinarystrihnspecifiesasubsetbysayianthestringihepresehedingaiiiion。Forexample，ifrings0111and0000staivelyfor{a2，a3，a4}，

ayset。Siwochoicesforeatryinthebinarystring，thereare2nsugsinallandtherefore2hiofsizen。

umbers

&hesimplestvisualrepresentationsthatgivesrisetothisypeisasthenumberofwayswedraw‘mountains’usingnupstrokesandndownstrokes（seeFigure3）

3。Withthreeupanddowherearefivemountainpatterns

&ainpatternhasaion，however，asameaningfulbradsothenumberofmeaningfulwaysaofnpairsofparehenthumber。Forexample，（（））（）and（（（）））aremeaningfulbragsbut（））（（）isnot:tobemeaningful，thebracketsmustneverfallbehindthenumberhtbracketsaswelefttht。Thisdstothenaturalthatourmountainsmustneverdiveunderground。ForihefirstandlastmountainpatternsinFigure3dt（）（（））aively。

&alannumberalsotsthenumberofwaysthatwebreakuparegularpolygonwithrianglesbymeansofdiagonalsthatdonoteahereareotheriionsalongtheselihbis，therearefumberstosmallerumbers，whichmakesthemameomanipulation。

Fibonaumbers

TheFibonaeseriesofengeionamongthegeneralpublic。Thesequensasfollows

1，1，2，3，5，8，13，21，34，55，89，144，233，377，610，···

whereeaumberafterthepairofinitial1sisthesumofthetwothatebefore。Inthis，thereisasimilaritywiththebisiermisthesumoftwopreviousohesequehemethodofformationoftheFibonaumbersissimpler:

fn=fn-1+festhenthFibonaumberandwefixf1=f2=1。Wecallsuulathatdefineseachmemberofasequespredecessorsareorarecerelation。

Howdoesthissequewasfirstintrodu1202byLeonardoofPisa，betterknownasFibonatheformofhiscelebratedRabbitProblem。Afemalerabbitisborwomourityaergivesbirthtoadaughtereath。ThenumberoffemalerabbitswehaveatthebegihistheheFibonaumbers，forthereis1rabbitatthebeginnimonth，aatthestartofthethirdmonthshegivesbirthtoadaughtersowethes。hshehasan3ahafterthatwehave5buhmotheradaughterareehebegihthereafter，thenumberhtersequalsthenumberoffemaleswehadtwomonthsago，asonlytheyareoldenoughtobreed。Itfollowsthatthenumberoffemaleswehaveatthestartofeachsubsequehetotalofthepreviousmonth（Fibonacci’srabbitsareimmortal）plusthehemohereforetheruleofformationoftheFibonaumbersexactlymatchesthebreedingpatternofhisrabbits。