Concrete mathematics : a foundation for computer science

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CONCRETE MATHEMATICS Dedicated to Leonhard Euler (1707-l 783)
CONCRETE MATHEMATICS Ronald L. Graham AT&T Bell Laboratories Donald E. Knuth Stanford University Oren Patashnik Stanford University A ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan
Page 1: CONCRETE MATHEMATICS Dedicated to L
Page 5 and 6: “A odience, level, and treatment
Page 7 and 8: ‘I a concrete life preserver thro
Page 9 and 10: I’m unaccustomed to this face. De
Page 11 and 12: n [ n-l 1 n {I m Prestressed concre
Page 13: CONTENTS xiii 5.3 Tricks of the Tra
Page 16 and 17: 2 RECURRENT PROBLEMS It’s not imm
Page 18 and 19: 4 RECURRENT PROBLEMS Of course the
Page 20 and 21: 6 RECURRENT PROBLEMS through any of
Page 22 and 23: 8 RECURRENT PROBLEMS From these sma
Page 24 and 25: 10 RECURRENT PROBLEMS But what abou
Page 26 and 27: 12 RECURRENT PROBLEMS that is, in t
Page 28 and 29: 14 RECURRENT PROBLEMS by separating
Page 30 and 31: 16 RECURRENT PROBLEMS For example,
Page 32 and 33: 18 RECURRENT PROBLEMS Homework exer
Page 34 and 35: 20 RECURRENT PROBLEMS 21 Suppose th
Page 36 and 37: 22 SUMS The three-dots notation has
Page 38 and 39: 24 SUMS A slightly modified form of
Page 40 and 41: 26 SUMS where A(n), B(n), and C(n)
Page 42 and 43: 28 SUMS Let’s apply these ideas t
Page 44 and 45: 30 SUMS 2.3 MANIPULATION OF SUMS No
Page 46 and 47: 32 SUMS Typically we use rule (2.20
Page 48 and 49: 34 SUMS because the derivative of a
Page 50 and 51: 36 SUMS instead of summing over all
Page 52 and 53:
38 SUMS The second sum here is zero
Page 54 and 55:
40 SUMS The normal way to evaluate
Page 56 and 57:
42 SUMS First, as usual, we look at
Page 58 and 59:
44 SUMS order to get an equation fo
Page 60 and 61:
46 SUMS One way to use this fact is
Page 62 and 63:
48 SUMS definition where the factor
Page 64 and 65:
50 SUMS Let’s try to recap this;
Page 66 and 67:
52 SUMS we need an appropriate defi
Page 68 and 69:
54 SUMS This formula indicates why
Page 70 and 71:
56 SUMS u(x) = x and Av(x) = 2’;
Page 72 and 73:
58 SUMS The definition in the previ
Page 74 and 75:
60 SUMS In fact, our definition of
Page 76 and 77:
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
Page 82 and 83:
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
Page 88 and 89:
74 INTEGER FUNCTIONS By the way, th
Page 90 and 91:
76 INTEGER FUNCTIONS (Previously K
Page 92 and 93:
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
Page 114 and 115:
100 INTEGER FUNCTIONS 43 Find an in
Page 116 and 117:
4 Number Theory INTEGERS ARE CENTRA
Page 118 and 119:
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
Page 130 and 131:
116 NUMBER THEORY A fraction m/n is
Page 132 and 133:
118 NUMBER THEORY A mediant fractio
Page 134 and 135:
120 NUMBER THEORY This representati
Page 136 and 137:
122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
Page 140 and 141:
126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
Page 144 and 145:
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
Page 150 and 151:
136 NUMBER THEORY Conversely, if f(
Page 152 and 153:
138 NUMBER THEORY (We must add 1 to
Page 154 and 155:
140 NUMBER THEORY The problem of co
Page 156 and 157:
142 NUMBER THEORY n is odd, n4 and
Page 158 and 159:
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
Page 164 and 165:
150 NUMBER THEORY 57 Let S(m,n) be
Page 166 and 167:
152 NUMBER THEORY 73 If the 0(n) +
Page 168 and 169:
154 BINOMIAL COEFFICIENTS For examp
Page 170 and 171:
156 BINOMIAL COEFFICIENTS And now t
Page 172 and 173:
158 BINOMIAL COEFFICIENTS property
Page 174 and 175:
160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
Page 178 and 179:
164 BINOMIAL COEFFICIENTS Table 164
Page 180 and 181:
166 BINOMIAL COEFFICIENTS not rows.
Page 182 and 183:
168 BINOMIAL COEFFICIENTS Equation
Page 184 and 185:
170 BINOMIAL COEFFICIENTS that m =
Page 186 and 187:
172 BINOMIAL COEFFICIENTS ifweuse (
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Page 190 and 191:
176 BINOMIAL COEFFICIENTS sum on th
Page 192 and 193:
178 BINOMIAL COEFFICIENTS some data
Page 194 and 195:
180 BINOMIAL COEFFICIENTS Problem 4
Page 196 and 197:
182 BINOMIAL COEFFICIENTS We’re l
Page 198 and 199:
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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Page 218 and 219:
204 BINOMIAL COEFFICIENTS This hold
Page 220 and 221:
206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
Page 224 and 225:
210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
Page 228 and 229:
214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
Page 232 and 233:
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
Page 240 and 241:
226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
Page 244 and 245:
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
Page 254 and 255:
240 BINOMIAL COEFFICIENTS 86 Let al
Page 256 and 257:
242 BINOMIAL COEFFICIENTS Research
Page 258 and 259:
244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
Page 262 and 263:
248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
Page 266 and 267:
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
Page 272 and 273:
258 SPECIAL NUMBERS Table 258 Stirl
Page 274 and 275:
260 SPECIAL NUMBERS ’ ’ ’ (Th
Page 276 and 277:
262 SPECIAL NUMBERS we found that a
Page 278 and 279:
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 =
Page 294 and 295:
280 SPECIAL NUMBERS The process of
Page 296 and 297:
282 SPECIAL NUMBERS Fk < n < Fk+l;
Page 298 and 299:
284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
Page 302 and 303:
288 SPECIAL NUMBERS The continuant
Page 304 and 305:
290 SPECIAL NUMBERS in four steps:
Page 306 and 307:
292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
Page 310 and 311:
296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
Page 314 and 315:
300 SPECIAL NUMBERS 44 Prove the co
Page 316 and 317:
302 SPECIAL NUMBERS 61 Prove the id
Page 318 and 319:
304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
Page 322 and 323:
308 GENERATING FUNCTIONS are added
Page 324 and 325:
310 GENERATING FUNCTIONS (The last
Page 326 and 327:
312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
Page 330 and 331:
316 GENERATING FUNCTIONS determine
Page 332 and 333:
318 GENERATING FUNCTIONS operation
Page 334 and 335:
320 GENERATING FUNCTIONS Table 320
Page 336 and 337:
322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
Page 340 and 341:
326 GENERATING FUNCTIONS Once we’
Page 342 and 343:
328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
Page 346 and 347:
332 GENERATING FUNCTIONS Finally, s
Page 348 and 349:
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
Page 354 and 355:
340 GENERATING FUNCTIONS F(z)' = i
Page 356 and 357:
342 GENERATING FUNCTIONS We can app
Page 358 and 359:
344 GENERATING FUNCTIONS Step 3 is
Page 360 and 361:
346 GENERATING FUNCTIONS Raney’s
Page 362 and 363:
348 GENERATING FUNCTIONS Notice tha
Page 364 and 365:
350 GENERATING FUNCTIONS Chapter 5
Page 366 and 367:
352 GENERATING FUNCTIONS Now let’
Page 368 and 369:
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
Page 374 and 375:
360 GENERATING FUNCTIONS 20 A power
Page 376 and 377:
362 GENERATING FUNCTIONS 29 What is
Page 378 and 379:
364 GENERATING FUNCTIONS 43 The New
Page 380 and 381:
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
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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
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
Page 400 and 401:
386 DISCRETE PROBABILITY where Ug i
Page 402 and 403:
388 DISCRETE PROBABILITY Repeating
Page 404 and 405:
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
Page 416 and 417:
402 DISCRETE PROBABILITY Let sj be
Page 418 and 419:
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
Page 438 and 439:
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
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
Page 462 and 463:
448 ASYMPTOTICS We might as well do
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450 ASYMPTOTIC3 a rough estimate an
Page 466 and 467:
452 ASYMPTOTICS This last estimate
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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(
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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
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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
Page 496 and 497:
482 ASYMPTOTIC3 Research problems 6
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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.
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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)
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510 ANSWERS TO EXERCISES In both ca
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512 ANSWERS TO EXERCISES 5.7 Yes, b
Page 528 and 529:
514 ANSWERS TO EXERCISES because al
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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
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522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539:
524 ANSWERS TO EXERCISES Since ~06j
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526 ANSWERS TO EXERCISJES 5.94 This
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528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545:
530 ANSWERS TO EXERCISIES 6.40 If 6
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532 ANSWERS TO EXERCISES solutions
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534 ANSWERS TO EXERCISES 6.62 (a) A
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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,+
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546 ANSWERS TO EXERCISES 7.40 The e
Page 562 and 563:
548 ANSWERS TO EXERCISES 1 + a. Hen
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550 ANSWERS TO EXERCISES applied to
Page 566 and 567:
552 ANSWERS TO EXERCISES 8.13 (Solu
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554 ANSWERS TO EXERCISES 8.24 (a) A
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556 ANSWERS TO EXERCISES 8.32 By sy
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558 ANSWERS TO EXERCISES 8.37 The n
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560 ANSWERS TO EXERCISES probabilit
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562 ANSWERS TO EXERCISES And since
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564 ANSWERS TO EXERCISES 8.56 If m
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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
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572 ANSWERS TO EXERCISES 9.42 The h
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574 ANSWERS TO EXERCISIES length of
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576 ANSWERS TO EXERCISES 9.58 Let 0
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B Bibliography HERE ARE THE WORKS c
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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,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
Page 604 and 605:
590 BIBLIOGRAPHY 160 K. Inkeri, “
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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
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598 BIBLIOGRAPHY 271 A. D. Solov’
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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
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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.
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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:
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