Concrete mathematics : a foundation for computer science

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592 BIBLIOGRAPHY 188 E.E. Kummer, “Uber die Erganzungssatze zu den allgemeinen Re- 603. ciprocitatsgesetzen,” Journal fiir die reine und angewandte Mathematik 44 (1852), 93-146. Reprinted in his Collected Papers, volume 1, 485-538. 189 R. P. Kurshan and B. Gopinath, “Recursively generated periodic se- 487. quences,” Canadian Journal of Mathematics 26 (1974), 1356-1371. 190 Thomas Fantet de Lagny, Analyse g&&ale ou Methodes nouvelles pour 290. resoudre les problemes de tous les genres et de tous les degr6 2 l’infini. Published as volume 11. of Memoires de 1’AcadCmie Royale des Sciences, Paris, 1733. 191 J.-L. de la Grange [Lagrange], “Demonstration d’un theoreme nouveau 604. concernant les nombreis premiers,” Nouveaux Memoires de 1’AcadCmie royale des Sciences et Belles-Lettres de Berlin (1771), 125-137. Reprinted in his auvres, volume 3, 425-438. 192 J.-L. de la Grange [Lagrange], “Sur une nouvelle espece de calcul relatif 456. a la differentiation & a l’integration des quantites variables,” Nouveaux Memoires de 1’AcadCmie royale des Sciences et Belles-Lettres de Berlin (1772), 185-221. Reprinted in his Ckvres, volume 3, 441-476. 193 I. Lah, “Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung 603. in der mathematischen. Statistik,” Mitteilungsblatt fiir Mathematische Statistik 7 (1955), 2033212. 194 Edmund Landau, Handbuch der Lehre von der Verteilung der Prim- 434, 605. zahlen, two volumes. Teubner, Leipzig, 1909. 195 Edmund Landau, Vorlesungen iiber Zahlentheorie, three volumes. Hirzel, 603. Leipzig, 1927. 195’ P. S. de la Place [Laplace], “Memoire sur les approximations des Formules 452. qui sont fonctions de t&-grands nombres,” Memoires de 1’Academie royale des Sciences de Paris (1782), l-88. Reprinted in his CEuvres Completes 10, 207-291. 196 Adrien-Marie Legendre, Essai sur la The’orie des Nombres. Paris, 1798; 602. second edition, 1808. Third edition (retitled The’orie des Nombres, in two volumes), 1830; fourth edition, Blanchard, 1955. 197 D. H. Lehmer, “Tests <strong>for</strong> primality by the converse of Fermat’s theorem,” 602. Bulletin of the American Mathematical Society, series 2, 33 (1927), 327- 340. 198 D. H. Lehmer, “On Stern’s diatomic series,” American Mathematical 604. Monthly 36 (1929), 59-67. 199 D. H. Lehmer, “On Euler’s totient function,” Bulletin of the American 511. Mathematical Society, #series 2, 38 (1932), 745-751.
168. 281. 605. 602. 604. 603. 536. 602, 603, 604. 603. 278, 603. 1. 602. 487. 455. 140. 280, 604. B BIBLIOGRAPHY 593 200 G. W. Leibniz, letter to Johann Bernoulli (May 1695), in Leibnizens mathematische Schriften, volume 3, 174-179. 201 C. G. Lekkerkerker, “Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci,” Simon Stevin 29 (1952), 190-195. 201’Elliott H. Lieb, “Residual entropy of square ice,” Physical Review 162 (1967), 162-172. 202 B. F. Logan, “The recovery of orthogonal polynomials from a sum of squares,” SIAM Journal on Mathematical Analysis 21(1990), 1031-1050. 202’ B. F. Logan, “Polynomials related to the Stirling numbers,” AT&T Bell Laboratories internal technical memorandum, August 10, 1987. 203 Calvin T. Long and Verner E. Hoggatt, Jr., “Sets of binomial coefficients with equal products,” Fibonacci Quarterly 12 (1974), 71-79. 204 Sam Loyd, Cyclopedia of Puzzles. Franklin Bigelow Corporation, Morningside Press, New York, 1914. 205 E. Lucas, “Sur les rapports qui existent entre la theorie des nombres et le Calcul integral,” Comptes Rendus hebdomadaires des seances de I’AcadCmie des Sciences (Paris) 82 (1876), 1303-1305. 206 Edouard Lucas, “Sur les congruences des nombres euleriens et des coefficients differentiels des fonctions trigonometriques, suivant un module premier,” Bulletin de la SociCtC mathematique de France 6 (1878), 49-54. 207 Edouard Lucas, ThCorie des Nombres, volume 1. Gauthier-Villars, Paris, 1891. 208 Edouard Lucas, Recreations mathematiques, four volumes. Gauthier- Villars, Paris, 1891-1894. Reprinted by Albert Blanchard, Paris, 1960. (The Tower of Hanoi is discussed in volume 3, pages 55-59.) 209 R. C. Lyness, ‘Cycles,” The Mathematical Gazette 26 (1942), 62. 210 R. C. Lyness, “Cycles,” The Mathematical Gazette 29 (1945), 231-233. 211 Colin Maclaurin, Collected Letters, edited by Stella Mills. Shiva Publishing, Nantwich, Cheshire, 1982. 212 P. A. MacMahon, “Application of a theory of permutations in circular procession to the theory of numbers,” Proceedings of the London Mathematical Society 23 (1892), 305-313. 213 I^u. V. MatiiBsevich, “Diofantovost’ perechislimykh mnozhestv,” Doklady Akademii Nauk SSSR 191 (1970), 279-282. English translation, with amendments by the author, “Enumerable sets are diophantine,” Soviet Mathematics 11 (1970), 354-357.
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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
Page 194 and 195:
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
Page 276 and 277:
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
Page 372 and 373:
358 GENERATING FUNCTIONS 5 Find a g
Page 374 and 375:
360 GENERATING FUNCTIONS 20 A power
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362 GENERATING FUNCTIONS 29 What is
Page 378 and 379:
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
Page 384 and 385:
370 DISCRETE PROBABILITY about R if
Page 386 and 387:
372 DISCRETE PROBABILITY The mean o
Page 388 and 389:
374 DISCRETE PROBABILITY more sprea
Page 390 and 391:
376 DISCRETE PROBABILITY hence VS =
Page 392 and 393:
378 DISCRETE PROBABILITY And we can
Page 394 and 395:
380 DISCRETE PROBABILITY variance b
Page 396 and 397:
382 DISCRETE PROBABILITY (1 +z+...
Page 398 and 399:
384 DISCRETE PROBABILITY ~1 and ~2
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386 DISCRETE PROBABILITY where Ug i
Page 402 and 403:
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
Page 412 and 413:
398 DISCRETE PROBABILITY For exampl
Page 414 and 415:
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
Page 434 and 435:
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
Page 448 and 449:
434 ASYMPTOTICS inefficient “beca
Page 450 and 451:
436 ASYMPTOTICS 9.3 0 MANIPULATION
Page 452 and 453:
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
Page 460 and 461:
446 ASYMPTOTICS Problem 5: An infin
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448 ASYMPTOTICS We might as well do
Page 464 and 465:
450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467:
452 ASYMPTOTICS This last estimate
Page 468 and 469:
454 ASYMPTOTICS Here’s another ex
Page 470 and 471:
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
Page 482 and 483:
468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485:
470 ASYMPTOTICS For example, we hav
Page 486 and 487:
472 ASYMPTOTICS The final task seem
Page 488 and 489:
474 ASYMPTOTICS This is our approxi
Page 490 and 491:
476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493:
478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495:
480 ASYMPTOTICS a What is the asymp
Page 496 and 497:
482 ASYMPTOTIC3 Research problems 6
Page 498 and 499:
484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501:
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
Page 514 and 515:
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
Page 520 and 521:
506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523:
508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525:
510 ANSWERS TO EXERCISES In both ca
Page 526 and 527:
512 ANSWERS TO EXERCISES 5.7 Yes, b
Page 528 and 529:
514 ANSWERS TO EXERCISES because al
Page 530 and 531:
516 ANSWERS TO EXERCISES 5.44 Cance
Page 532 and 533:
518 ANSWERS TO EXERCISES and this i
Page 534 and 535:
520 ANSWERS TO EXERCISES The infini
Page 536 and 537:
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 604 and 605: 590 BIBLIOGRAPHY 160 K. Inkeri, “
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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