Concrete mathematics : a foundation for computer science

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76 INTEGER FUNCTIONS (Previously K was 10.) The total number of winners for general N comes to W = x (3k+4) +x[K3
. . . without MS of generality. . “If x be an incommensurable number less than unity, one of the series of quantities m/x, m/(1 -x), where m is a whole number, can be found which shall he between any given consecutive integers, and but one such quantity can be found.” - Rayleigh [245] Right, because exact/y one of the counts must increase when n increases by 1 . 3.2 FLOOR/CEILING APPLICATIONS 77 Our last application in this section looks at so-called spectra. We define the spectrum of a real number a to be an infinite multiset of integers, Sped4 = 114, 12a1, 13a1, . . .I. (A multiset is like a set but it can have repeated elements.) For example, the spectrum of l/2 starts out (0, 1, 1,2,2,3,3,. . .}. It’s easy to prove that no two spectra are equal-that a # (3 implies Spec(a) # Spec((3). For, assuming without loss of generality that a < (3, there’s a positive integer m such that m( l3 - a) 3 1. (In fact, any m 3 [l/( (3 - a)] will do; but we needn’t show off our knowledge of floors and ceilings all the time.) Hence ml3 - ma 3 1, and LrnSl > [ma]. Thus Spec((3) has fewer than m elements < lrnaj, while Spec(a) has at least m. Spectra have many beautiful properties. For example, consider the two multisets Spec(&) = {1,2,4,5,7,8,9,11,12,14,15,16,18,19,21,22,24 ,... }, Spec(2+fi) = {3,6,10,13,17,20,23,27,30,34,37,40,44,47,51,... }. It’s easy to calculate Spec( fi ) with a pocket calculator, and the nth element of Spec(2+ fi) is just 2n more than the nth element of Spec(fi), by (3.6). A closer look shows that these two spectra are also related in a much more surprising way: It seems that any number missing from one is in the other, but that no number is in both! And it’s true: The positive integers are the disjoint union of Spec( fi ) and Spec(2+ fi ). We say that these spectra form a partition of the positive integers. To prove this assertion, we will count how many of the elements of Spec(&!) are 6 n, and how many of the elements of Spec(2+fi) are 6 n. If the total is n, for each n, these two spectra do indeed partition the integers. Let a be positive. The number of elements in Spec(a) that are < n is N(a,n) = x[lkaJ O = x[[kaj O = tr kaO = x[O
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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
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
Page 78 and 79: 64 SUMS 18 Let 9%~ and Jz be the re
Page 80 and 81: 66 SUMS 34 35 36 Prove that if the
Page 82 and 83: 68 INTEGER FUNCTIONS below the line
Page 84 and 85: 70 INTEGER FUNCTIONS same inequalit
Page 86 and 87: 72 INTEGER FUNCTIONS An important s
Page 88 and 89: 74 INTEGER FUNCTIONS By the way, th
Page 92 and 93: 78 INTEGER FUNCTIONS This derivatio
Page 94 and 95: 80 INTEGER FUNCTIONS We’ve got mo
Page 96 and 97: 82 INTEGER FUNCTIONS division, and
Page 98 and 99: 84 INTEGER FUNCTIONS more than one
Page 100 and 101: 86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
Page 102 and 103: 88 INTEGER FUNCTIONS For this purpo
Page 104 and 105: 90 INTEGER FUNCTIONS it comes out i
Page 106 and 107: 92 INTEGER FUNCTIONS To summarize,
Page 108 and 109: 94 INTEGER FUNCTIONS Also, as we gu
Page 110 and 111: 96 INTEGER FUNCTIONS Basics 10 Show
Page 112 and 113: 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
Page 120 and 121: 106 NUMBER THEORY product of primes
Page 122 and 123: 108 NUMBER THEORY only finitely man
Page 124 and 125: 110 NUMBER THEORY But 2P - 1 isn’
Page 126 and 127: 112 NUMBER THEORY arrangements in a
Page 128 and 129: 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
Page 138 and 139: 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
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376 DISCRETE PROBABILITY hence VS =
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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
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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
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428 ASYMPTOTIC3 How about the funct
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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
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440 ASYMPTOTICS Notice how the 0 te
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442 ASYMPTOTICS Problem 3: The nth
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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
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458 ASYMPTOTICS (The Bernoulli poly
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460 ASYMPTOTICS The graph of B,(x)
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462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
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464 ASYMPTOTICS The integral JT B4(
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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
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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
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482 ASYMPTOTIC3 Research problems 6
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484 ANSWERS TO EXERCISES Venn [294]
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486 ANSWERS TO EXERCISES move the b
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488 ANSWERS TO EXERCISES 2.2 This i
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490 ANSWERS TO EXERCISES 2.24 Summi
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492 ANSWERS TO EXERCISES 2.36 (a) B
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494 ANSWERS TO EXERCISES 3.21 If 10
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496 ANSWERS TO EXERCISES the circle
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498 ANSWERS TO EXERCISES 3.40 Let L
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500 ANSWERS TO EXERCISES 3.50 H. S.
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502 ANSWERS TO EXERCISES 4.22 (b,
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504 ANSWERS TO EXERCISES 4.40 Let f
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506 ANSWERS TO EXERCISES 4.50 (a) I
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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
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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
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520 ANSWERS TO EXERCISES The infini
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522 ANSWERS TO EXERCISES 5.82 Let ~
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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
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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(
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538 ANSWERS TO EXERCISIES One of th
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540 ANSWERS TO EXERCISES 6.85 The p
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542 ANSWERS TO EXERCISES Hence gzn+
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544 ANSWERS TO EXERCISES 7.24 ntk,+
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546 ANSWERS TO EXERCISES 7.40 The e
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548 ANSWERS TO EXERCISES 1 + a. Hen
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550 ANSWERS TO EXERCISES applied to
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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
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568 ANSWERS TO EXERCISES 9.19 Hlo =
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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
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582 BIBLIOGRAPHY 51 52 53 54 55 56
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584 BIBLIOGRAPHY 78 79 80 81 82 83
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586 BIBLIOGRAPHY 100 R. A. Fisher,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
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590 BIBLIOGRAPHY 160 K. Inkeri, “
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592 BIBLIOGRAPHY 188 E.E. Kummer,
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594 BIBLIOGRAPHY 214 Z.A. Melzak, C
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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
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604 CREDITS FOR EXERCISES 6.46 6.47
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Index WHEN AN INDEX ENTRY refers to
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608 INDEX Bois-Reymond, Paul David
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610 INDEX Degenerate hypergeometric
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612 INDEX Feder, Tom&s, 604. Feigen
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614 INDEX Hall, Marshall, Jr., 588.
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616 INDEX Lincoln, Abraham, 387. Li
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618 INDEX Patashnik, Oren, iii, iv,
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620 INDEX Running time, 411-412. Ru
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622 INDEX Sylvester, James Joseph,
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List of Tables Sums and differences
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Concrete mathematics : a foundation for computer science

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