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1、CONCRETEMATHEMATICSSecondEditionDedicatedtoLeonhardEuler(1707{1783)AFoundationforComputerScienceCONCRETEMATHEMATICSSecondEditionRonaldL.GrahamAT&TBellLaboratoriesDonaldE.KnuthStanfordUniversityOrenPatashnikCenterforCommunicationsResearch677ADDISON-WESLEYPUBLISHIN
2、GCOMPANYReading,MassachusettsMenloPark,CaliforniaNewYorkDonMills,OntarioWokingham,EnglandAmsterdamBonnSydneySingaporeTokyoMadridSanJuanMilanParisLibraryofCongressCataloging-in-PublicationDataGraham,RonaldLewis,1935-Concretemathematics:afoundationforcomputerscienc
3、e/RonaldL.Graham,DonaldE.Knuth,OrenPatashnik.--2nded.xiii,657p.24cm.Bibliography:p.604Includesindex.ISBN0-201-55802-51.Mathematics.2.Computerscience--Mathematics.I.Knuth,DonaldErvin,1938-.II.Patashnik,Oren,1954-.III.Title.QA39.2.G7331994510--dc2093-40325CIPReprod
4、ucedbyAddison-Wesleyfromcamera-readycopysuppliedbytheauthors.Copyright°c1994,1989byAddison-WesleyPublishingCompany,Inc.Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechan-ical,pho
5、tocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.12345678910{MA{9897969594PrefaceAudience,level,THISBOOKISBASEDonacourseofthesamenamethathasbeentaughtandtreatment
6、annuallyatStanfordUniversitysince19
7、70.About ftystudentshavetakenitadescriptionofeachyear
8、juniorsandseniors,butmostlygraduatestudents
9、andalumnisuchmattersiswhatprefacesareoftheseclasseshavebeguntospawnsimilarcourseselsewhere.Thusthetimesupposedtobeseemsripetopresentthematerialtoawideraudience(inclu
10、dingsophomores).about."ItwasadarkandstormydecadewhenConcreteMathematicswasborn.
11、P.R.Halmos[173]Long-heldvalueswereconstantlybeingquestionedduringthoseturbulentyears;collegecampuseswerehotbedsofcontroversy.Thecollegecurriculumitselfwaschallenged,andmathematicsdidn
12、otescapescrutiny.JohnHam-mersleyhadjustwrittenathought-provokingarticleOntheenfeeblementofmathematicalskillsby`ModernMathematics'andbysimilarsoftintellectualt
