Concrete mathematics : a foundation for computer science

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590 BIBLIOGRAPHY 160 K. Inkeri, “Absch;itzu:ngen fiir eventuelle Lijsungen der Gleichung im 509. Fermatschen Problem,” Annales Universitatis Turkuensis, series A, 16, 1 (1953), 3-9. 161 Kenneth E. Iverson, A Programming Language. Wiley, 1962. 24, 67, 602. 162 C. G. J. Jacobi, Fundamenta nova theor& functionurn ellipticarum. 64. K6nigsberg, BorntrBger, 1829. Reprinted in his Gesammelte Werke, volume 1, 49-239. 163 Dov Jarden and Theodor Motzkin, “The product of sequences with a 533. common linear recursion <strong>for</strong>mula of order 2,” Riveon Lematematika 3 (1949), 25-27, 38 (Heb’rew with English summary). English version reprinted in Dov Jarden., Recurring Sequences, Jerusalem, 1958, 42-45; second edition, Jerusalem, 1966, 30-33. 164 Arne Jonassen and Donald E. Knuth, “A trivial algorithm whose analysis 520. isn’t,” Journal of Computer and System Sciences 16 (1978), 301-322. 165 Bush Jones, “Note on internal merging,” Software -Practice and Expe- 175. rience 2 (1972), 241-243. 166 Flavius Josephus, ETOPIA IOTAAIKOT ITOAEMOT ITPOC Pa- 8. MAIOTC. English translation, History of the Jewish War against the Remans, by H. St. J. Thackeray, in the Loeb Classical Library edition of Josephus’s works, volumes 2 and 3, Heinemann, London, 1927-1928. (The “Josephus problem” may be based on an early manuscript now preserved only in the Slavonic version; see volume 2, page xi, and volume 3, page 654.) 167 R. Jungen, “Sur les series de Taylor n’ayant que des singularit& 604. algCbrico-logarithmiques sur leur cercle de convergence,” Commentarii Mathematici Helvetici 3 (1931), 266-306. 168 I. Kauckp, “Problem 132257: A harmonic identity,” American Mathe- 604. matical Monthly 78 (1971), 908. 169 Murray S. Klamkin, Initernational Mathematical Olympiads, 1978-1985, 602, 603. and Forty Supplementary Problems. Mathematical Association of America, 1986. 170 Konrad Knopp, Theorie und Anwendung der unendlichen Reihen. Julius 605. Springer, Berlin, 1922; second edition, 1924. Reprinted by Dover, 1945. Fourth edition, 1947; fifth edition, 1964. English translation, Theory and Application of Infinite Series, 1928; second edition, 1951. 171 Donald E. Knuth, “Euler’s constant to 1271 places,” Mathematics of 467. Computation 16 (1962), 275-281. 172 Donald Knuth, “Transcendental numbers based on the Fibonacci se- 531. quence,” Fibonacci Quarterly 2 (1964), 43-44, 52.
vi, 486, 499, 515, 548, 602, 603, 604, 605. 110, 128, 486, 602, 604, 605. 253, 397, 487, 603, 604, 605. 603. 605. 602. 540. 602. 602. 603. 604. 538. 532. 111. 213, 603. B BIBLIOGRAPHY 591 173 Donald E. Knuth, The Art of Computer Programming, volume 1: Fundamental Algor.ithms. Addison-Wesley, 1968; second edition, 1973. 174 Donald E. Knuth, The Art of Computer Programming, volume 2: Seminumerical Algorithms. Addison-Wesley, 1969; second edition, 1981. 175 Donald E. Knuth, The Art of Computer Programming, volume 3: Sorting and Searching. Addison-Wesley, 1973; second printing, 1975. 176 Donald E. Knuth, “Problem E 2492: Some sum,” American Mathematical Monthly 82 (1975), 855. 177 Donald E. Knuth, Mariages stables et leurs relations avec d’autres probl&mes combinatoires. Les Presses de 1’UniversitC de Mont&al, 1976. Revised and corrected edition, 1980. 178 Donald E. Knuth, The wbook. Addison-Wesley, 1984. Reprinted as volume A of Computers & Typesetting, 1986. 179 Donald E. Knuth, “An analysis of optimum caching,” Journal of Algorithms 6 (1985), 181-199. 180 Donald E. Knuth, Computers & Typesetting, volume D: METRFONT: The Program. Addison-Wesley, 1986. 181 Donald E. Knuth, “Problem 1280,” Mathematics Magazine 60 (1987), 329. 182 Donald E. Knuth, “Problem E 3106: A new sum <strong>for</strong> n’,” American Mathematical Monthly 94 (1987), 795-797. 183 Donald E. Knuth, “Fibonacci multiplication,” Applied Mathematics Letters 1 (1988), 57-60. 184 Donald E. Knuth, “A Fibonacci-like sequence of composite numbers,” Mathematics Magazine 63 (1990), 21-25. 185 Donald E. Knufh and Thomas J. Buckholtz, “Computation of Tangent, Euler, and Bernoulli numbers,” Mathematics of Computation 21 (1967), 663-688. 186 C. Kramp, lhmens d’arithmgtique universelle. Cologne, 1808. 187 E. E. Kummer, “Ueber die hypergeometrische Reihe , I 4 x+m+wB+li 1 .Y 1.2.y(y+l) xx +a(a+l)(a+2)8(8+1)(B+2)X3+ 1.2.3.y(y+l)(y+2) "' In Journal fiir die reine und angewandte Mathematik 15 (1836), 39-83, 127-172. Reprinted in his Collected Papers, volume 2, 75-166.
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CONCRETE MATHEMATICS Dedicated to L
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CONCRETE MATHEMATICS Ronald L. Grah
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“A odience, level, and treatment
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‘I a concrete life preserver thro
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I’m unaccustomed to this face. De
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n [ n-l 1 n {I m Prestressed concre
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CONTENTS xiii 5.3 Tricks of the Tra
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2 RECURRENT PROBLEMS It’s not imm
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4 RECURRENT PROBLEMS Of course the
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6 RECURRENT PROBLEMS through any of
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8 RECURRENT PROBLEMS From these sma
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10 RECURRENT PROBLEMS But what abou
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12 RECURRENT PROBLEMS that is, in t
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14 RECURRENT PROBLEMS by separating
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16 RECURRENT PROBLEMS For example,
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18 RECURRENT PROBLEMS Homework exer
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20 RECURRENT PROBLEMS 21 Suppose th
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22 SUMS The three-dots notation has
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24 SUMS A slightly modified form of
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26 SUMS where A(n), B(n), and C(n)
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28 SUMS Let’s apply these ideas t
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30 SUMS 2.3 MANIPULATION OF SUMS No
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32 SUMS Typically we use rule (2.20
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34 SUMS because the derivative of a
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36 SUMS instead of summing over all
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38 SUMS The second sum here is zero
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40 SUMS The normal way to evaluate
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42 SUMS First, as usual, we look at
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44 SUMS order to get an equation fo
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46 SUMS One way to use this fact is
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48 SUMS definition where the factor
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50 SUMS Let’s try to recap this;
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52 SUMS we need an appropriate defi
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54 SUMS This formula indicates why
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56 SUMS u(x) = x and Av(x) = 2’;
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58 SUMS The definition in the previ
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60 SUMS In fact, our definition of
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62 SUMS tCj,kiEF a. J, k < A for al
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64 SUMS 18 Let 9%~ and Jz be the re
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66 SUMS 34 35 36 Prove that if the
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68 INTEGER FUNCTIONS below the line
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70 INTEGER FUNCTIONS same inequalit
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72 INTEGER FUNCTIONS An important s
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74 INTEGER FUNCTIONS By the way, th
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76 INTEGER FUNCTIONS (Previously K
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78 INTEGER FUNCTIONS This derivatio
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80 INTEGER FUNCTIONS We’ve got mo
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82 INTEGER FUNCTIONS division, and
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84 INTEGER FUNCTIONS more than one
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86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
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88 INTEGER FUNCTIONS For this purpo
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90 INTEGER FUNCTIONS it comes out i
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92 INTEGER FUNCTIONS To summarize,
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94 INTEGER FUNCTIONS Also, as we gu
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96 INTEGER FUNCTIONS Basics 10 Show
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98 INTEGER FUNCTIONS 31 Prove or di
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100 INTEGER FUNCTIONS 43 Find an in
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4 Number Theory INTEGERS ARE CENTRA
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104 NUMBER THEORY The left side can
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106 NUMBER THEORY product of primes
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108 NUMBER THEORY only finitely man
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110 NUMBER THEORY But 2P - 1 isn’
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112 NUMBER THEORY arrangements in a
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114 NUMBER THEORY The binary repres
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116 NUMBER THEORY A fraction m/n is
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118 NUMBER THEORY A mediant fractio
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120 NUMBER THEORY This representati
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122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
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126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
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130 NUMBER THEORY Now we must show
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132 NUMBER THEORY And it’s possib
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134 NUMBER THEORY For example, (~(1
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136 NUMBER THEORY Conversely, if f(
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138 NUMBER THEORY (We must add 1 to
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140 NUMBER THEORY The problem of co
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142 NUMBER THEORY n is odd, n4 and
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144 NUMBER THEORY But the inner sum
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146 NUMBER THEORY 22 The number 111
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148 NUMBER THEORY 40 If the radix p
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150 NUMBER THEORY 57 Let S(m,n) be
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152 NUMBER THEORY 73 If the 0(n) +
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154 BINOMIAL COEFFICIENTS For examp
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156 BINOMIAL COEFFICIENTS And now t
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158 BINOMIAL COEFFICIENTS property
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160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
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164 BINOMIAL COEFFICIENTS Table 164
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166 BINOMIAL COEFFICIENTS not rows.
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168 BINOMIAL COEFFICIENTS Equation
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170 BINOMIAL COEFFICIENTS that m =
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172 BINOMIAL COEFFICIENTS ifweuse (
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176 BINOMIAL COEFFICIENTS sum on th
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178 BINOMIAL COEFFICIENTS some data
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180 BINOMIAL COEFFICIENTS Problem 4
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182 BINOMIAL COEFFICIENTS We’re l
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184 BINOMIAL COEFFICIENTS We can’
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186 BINOMIAL COEFFICIENTS 5.3 TRICK
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188 BINOMIAL COEFFICIENTS if we app
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190 BINOMIAL COEFFICIENTS The Newto
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192 BINOMIAL COEFFICIENTS One examp
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194 BINOMIAL COEFFICIENTS We can de
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196 BINOMIAL COEFFICIENTS This obse
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198 BINOMIAL COEFFICIENTS Generatin
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200 BINOMIAL COEFFICIENTS can be pu
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204 BINOMIAL COEFFICIENTS This hold
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206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
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210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
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214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
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218 BINOMIAL COEFFICIENTS It’s al
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220 BINOMIAL COEFFICIENTS A similar
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222 BINOMIAL COEFFICIENTS One use o
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224 BINOMIAL COEFFICIENTS Therefore
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226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
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230 BINOMIAL COEFFICIENTS valid for
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232 BINOMIAL COEFFICIENTS 18 Find a
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234 BINOMIAL COEFFICIENTS 37 Show t
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236 BINOMIAL COEFFICIENTS 57 Use Go
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238 BINOMIAL COEFFICIENTS 72 Prove
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240 BINOMIAL COEFFICIENTS 86 Let al
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242 BINOMIAL COEFFICIENTS Research
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244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
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248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
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252 SPECIAL NUMBERS We can remember
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254 SPECIAL NUMBERS Table 254 Euler
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{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
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258 SPECIAL NUMBERS Table 258 Stirl
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260 SPECIAL NUMBERS ’ ’ ’ (Th
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262 SPECIAL NUMBERS we found that a
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264 SPECIAL NUMBERS Rearranging, we
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266 SPECIAL NUMBERS Thus we have th
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268 SPECIAL NUMBERS (We have used t
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270 SPECIAL NUMBERS Bernoulli numbe
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272 SPECIAL NUMBERS trigonometric f
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274 SPECIAL NUMBERS Recurrence (6.9
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276 SPECIAL NUMBERS Gnethingwecando
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278 SPECIAL NUMBERS cases are: a0 =
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280 SPECIAL NUMBERS The process of
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282 SPECIAL NUMBERS Fk < n < Fk+l;
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284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
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288 SPECIAL NUMBERS The continuant
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290 SPECIAL NUMBERS in four steps:
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292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
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296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
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300 SPECIAL NUMBERS 44 Prove the co
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302 SPECIAL NUMBERS 61 Prove the id
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304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
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308 GENERATING FUNCTIONS are added
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310 GENERATING FUNCTIONS (The last
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312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
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316 GENERATING FUNCTIONS determine
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318 GENERATING FUNCTIONS operation
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320 GENERATING FUNCTIONS Table 320
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322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
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326 GENERATING FUNCTIONS Once we’
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328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
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332 GENERATING FUNCTIONS Finally, s
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334 GENERATING FUNCTIONS Step 3 was
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336 GENERATING FUNCTIONS This is a
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338 GENERATING FUNCTIONS Table 337
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340 GENERATING FUNCTIONS F(z)' = i
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342 GENERATING FUNCTIONS We can app
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344 GENERATING FUNCTIONS Step 3 is
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346 GENERATING FUNCTIONS Raney’s
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348 GENERATING FUNCTIONS Notice tha
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350 GENERATING FUNCTIONS Chapter 5
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352 GENERATING FUNCTIONS Now let’
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354 GENERATING FUNCTIONS function f
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356 GENERATING FUNCTIONS 7.7 DIRICH
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358 GENERATING FUNCTIONS 5 Find a g
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360 GENERATING FUNCTIONS 20 A power
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362 GENERATING FUNCTIONS 29 What is
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364 GENERATING FUNCTIONS 43 The New
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366 GENERATING FUNCTIONS 53 The seq
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368 DISCRETE PROBABILITY total of 6
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370 DISCRETE PROBABILITY about R if
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372 DISCRETE PROBABILITY The mean o
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374 DISCRETE PROBABILITY more sprea
Page 390 and 391:
376 DISCRETE PROBABILITY hence VS =
Page 392 and 393:
378 DISCRETE PROBABILITY And we can
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380 DISCRETE PROBABILITY variance b
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382 DISCRETE PROBABILITY (1 +z+...
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384 DISCRETE PROBABILITY ~1 and ~2
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386 DISCRETE PROBABILITY where Ug i
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388 DISCRETE PROBABILITY Repeating
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390 DISCRETE PROBABILITY There’s
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392 DISCRETE PROBABILITY But the co
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394 DISCRETE PROBABILITY In the spe
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396 DISCRETE PROBABILITY Here 1 is
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398 DISCRETE PROBABILITY For exampl
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400 DISCRETE PROBABILITY Thus the m
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402 DISCRETE PROBABILITY Let sj be
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404 DISCRETE PROBABILITY Therefore
Page 420 and 421:
406 DISCRETE PROBABILITY The mean v
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408 DISCRETE PROBABILITY Complicate
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410 DISCRETE PROBABILITY all k and
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412 DISCRETE PROBABILITY to be a pg
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414 DISCRETE PROBABILITY 10 What’
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416 DISCRETE PROBABILITY 28 What is
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418 DISCRETE PROBABILITY 38 What is
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420 DISCRETE PROBABIL1T.Y a Once he
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422 DISCRETE PROBABILITY 50 Conside
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424 DISCRETE PROBABILITY 58 Are the
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426 ASYMPTOTICS The word asymptotic
Page 442 and 443:
428 ASYMPTOTIC3 How about the funct
Page 444 and 445:
430 ASYMPTOTICS N. G. de Bruijn beg
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432 ASYMPTOTICS subtle consideratio
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434 ASYMPTOTICS inefficient “beca
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436 ASYMPTOTICS 9.3 0 MANIPULATION
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138 ASYMPTOTICS Table 438 Asymptoti
Page 454 and 455:
440 ASYMPTOTICS Notice how the 0 te
Page 456 and 457:
442 ASYMPTOTICS Problem 3: The nth
Page 458 and 459:
444 ASYMPTOTICS Problem 4: A sum fr
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446 ASYMPTOTICS Problem 5: An infin
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448 ASYMPTOTICS We might as well do
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450 ASYMPTOTIC3 a rough estimate an
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452 ASYMPTOTICS This last estimate
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454 ASYMPTOTICS Here’s another ex
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456 ASYMPTOTICS On the left is a ty
Page 472 and 473:
458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475:
460 ASYMPTOTICS The graph of B,(x)
Page 476 and 477:
462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
Page 478 and 479:
464 ASYMPTOTICS The integral JT B4(
Page 480 and 481:
466 ASYMPTOTIC3 where 0 < a,,,, < 1
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468 ASYMPTOTICS In Chapter 5, equat
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470 ASYMPTOTICS For example, we hav
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472 ASYMPTOTICS The final task seem
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474 ASYMPTOTICS This is our approxi
Page 490 and 491:
476 ASYMPTOTIC23 9 Prove (9.22) rig
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478 ASYMPTOTICS 39 Evaluate xOsk
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480 ASYMPTOTICS a What is the asymp
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482 ASYMPTOTIC3 Research problems 6
Page 498 and 499:
484 ANSWERS TO EXERCISES Venn [294]
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486 ANSWERS TO EXERCISES move the b
Page 502 and 503:
488 ANSWERS TO EXERCISES 2.2 This i
Page 504 and 505:
490 ANSWERS TO EXERCISES 2.24 Summi
Page 506 and 507:
492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509:
494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511:
496 ANSWERS TO EXERCISES the circle
Page 512 and 513:
498 ANSWERS TO EXERCISES 3.40 Let L
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500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517:
502 ANSWERS TO EXERCISES 4.22 (b,
Page 518 and 519:
504 ANSWERS TO EXERCISES 4.40 Let f
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506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523:
508 ANSWERS TO EXERCISES and (k- 1)
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510 ANSWERS TO EXERCISES In both ca
Page 526 and 527:
512 ANSWERS TO EXERCISES 5.7 Yes, b
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514 ANSWERS TO EXERCISES because al
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516 ANSWERS TO EXERCISES 5.44 Cance
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518 ANSWERS TO EXERCISES and this i
Page 534 and 535:
520 ANSWERS TO EXERCISES The infini
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522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539:
524 ANSWERS TO EXERCISES Since ~06j
Page 540 and 541:
526 ANSWERS TO EXERCISJES 5.94 This
Page 542 and 543:
528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545:
530 ANSWERS TO EXERCISIES 6.40 If 6
Page 546 and 547:
532 ANSWERS TO EXERCISES solutions
Page 548 and 549:
534 ANSWERS TO EXERCISES 6.62 (a) A
Page 550 and 551:
536 ANSWERS TO EXERCISES 6.74 If p(
Page 552 and 553:
538 ANSWERS TO EXERCISIES One of th
Page 554 and 555: 540 ANSWERS TO EXERCISES 6.85 The p
Page 556 and 557: 542 ANSWERS TO EXERCISES Hence gzn+
Page 558 and 559: 544 ANSWERS TO EXERCISES 7.24 ntk,+
Page 560 and 561: 546 ANSWERS TO EXERCISES 7.40 The e
Page 562 and 563: 548 ANSWERS TO EXERCISES 1 + a. Hen
Page 564 and 565: 550 ANSWERS TO EXERCISES applied to
Page 566 and 567: 552 ANSWERS TO EXERCISES 8.13 (Solu
Page 568 and 569: 554 ANSWERS TO EXERCISES 8.24 (a) A
Page 570 and 571: 556 ANSWERS TO EXERCISES 8.32 By sy
Page 572 and 573: 558 ANSWERS TO EXERCISES 8.37 The n
Page 574 and 575: 560 ANSWERS TO EXERCISES probabilit
Page 576 and 577: 562 ANSWERS TO EXERCISES And since
Page 578 and 579: 564 ANSWERS TO EXERCISES 8.56 If m
Page 580 and 581: 566 ANSWERS TO EXERCISES We can now
Page 582 and 583: 568 ANSWERS TO EXERCISES 9.19 Hlo =
Page 584 and 585: 570 ANSWERS TO EXERCIS:ES 9.30 Let
Page 586 and 587: 572 ANSWERS TO EXERCISES 9.42 The h
Page 588 and 589: 574 ANSWERS TO EXERCISIES length of
Page 590 and 591: 576 ANSWERS TO EXERCISES 9.58 Let 0
Page 592 and 593: B Bibliography HERE ARE THE WORKS c
Page 594 and 595: 580 BIBLIOGRAPHY 24' 25 26 27 28 29
Page 596 and 597: 582 BIBLIOGRAPHY 51 52 53 54 55 56
Page 598 and 599: 584 BIBLIOGRAPHY 78 79 80 81 82 83
Page 600 and 601: 586 BIBLIOGRAPHY 100 R. A. Fisher,
Page 602 and 603: 588 BIBLIOGRAPHY 128 Ronald L. Grah
Page 606 and 607: 592 BIBLIOGRAPHY 188 E.E. Kummer,
Page 608 and 609: 594 BIBLIOGRAPHY 214 Z.A. Melzak, C
Page 610 and 611: 596 BIBLIOGRAPHY 243 George N. Rane
Page 612 and 613: 598 BIBLIOGRAPHY 271 A. D. Solov’
Page 614 and 615: 600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617: 602 CREDITS FOR EXERCISES 1.1 1.2 1
Page 618 and 619: 604 CREDITS FOR EXERCISES 6.46 6.47
Page 620 and 621: Index WHEN AN INDEX ENTRY refers to
Page 622 and 623: 608 INDEX Bois-Reymond, Paul David
Page 624 and 625: 610 INDEX Degenerate hypergeometric
Page 626 and 627: 612 INDEX Feder, Tom&s, 604. Feigen
Page 628 and 629: 614 INDEX Hall, Marshall, Jr., 588.
Page 630 and 631: 616 INDEX Lincoln, Abraham, 387. Li
Page 632 and 633: 618 INDEX Patashnik, Oren, iii, iv,
Page 634 and 635: 620 INDEX Running time, 411-412. Ru
Page 636 and 637: 622 INDEX Sylvester, James Joseph,
Page 638 and 639: List of Tables Sums and differences
show all

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