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Chapter 5 Numbers that count（第1页）

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&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.