Artigo de 1990 de Richard Guy Sobre a Lei Forte Dos Pequenos Numeros

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The Second Strong Law of Small NumbersAuthor(s): Richard K. GuyReviewed work(s):Source: Mathematics Magazine, Vol. 63, No. 1 (Feb., 1990), pp. 3-20Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2691503 .

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A RT C LE S

The Second StrongLawofSmall Numbers

RICHARD K. GUYThe Universityf CalgaryAlberta, anada T2N1N4

You have probably lreadymet The Strong aw ofSmall Numbers, ither ormally[15, 21, 22]

There ren't nough mallnumbers o meet hemanydemandsmade ofthem

or in some frustratednd semiconscious ormulationhat occurred o you in therough-and-tumblef everydaymathematicalnquiry. t is the constant nemyofmathematical iscovery:t once theScylla, hatteringensible tatement ith puri-ous exceptions,nd theCharybdisfcapricious oincidences,ausing areless onjec-tures: the dilemma to searchforproofor forcounterexample.t fooledFermat(Example 1 of [21]) and we'll meet Euler'smemorablexample t the end of thearticle.

It's timeto introduce he SecondStrong aw ofSmallNumbers:

When wonumbersook qual,it

ain't necessarilyo!

"How can thispossibly e?" I hearyouask.By wayofanswer invite outo examine

Example 36 Evaluate the polynomial n4 6n3+ 24n2 18n+ 24)/24 for n=

1,2,3, ..Examples1 to 35 are n [21];there ollow orty-fourore. n each, youare invited

toguesswhatpatternf numberss emerging,ndtodecide whetherhepatternwillpersist.Manyof theexamples refraudulent,utsomegenuine heoremsremingledin,tokeep youonyour oes, nd theremayeven be an unsolved roblem r two.

Examples37 to40 involve ascal'striangle.

Example37Pascal's trianglemodulo ) has beena perennial opic.But haveyou tried eading

the rows s binarynumbers? , 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285,3855, 4369,13107, 21845,65535,65537,... . Rememberhat here re zerosoutside hetriangleas well, so you can also include theirdoubles, 2, 6, 10, 30, 34, 102,..., theirquadruples, , 12, 20, 60, 68,..., and so on, as well, fyoulike. Do you recognizethese numbers?





VOL. 63, NO. 1, FEBRUARY 1990 5

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 15 1 5 10 10 5 1

6 1 6 15 20 15 6 1

7 1 7 21 35 35 21 7 1

8 1 8 28 56 17 56 28 8 1

9 1 9 36 84 126 126 84 36 9 1

10 1 10 45 120 210 252 210 120 45 10 1

11 1 11 55 165 330 462 462 330 165 55 11 1

12 1 12 66 220 495 792 924 792 495 220 66 12 1

Whatare therow numbersprinted n bold) of the rowswhich ontain t least onebold entry: ,6,8,9,10,12,...?

Example40 We finally rawPascal's triangle ith n) in the nthrow nd (n + r)thcolumn:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160 11 11

2 1 2 13 1 3 3 14 1 4 6 4 15 1 5 10 10 5 16 1 6 15 20 15 6 17 1 7 21 35 35 21 7 18 1 8 28 56 70 56 28 8 19 1 9 36 84 126 126 84 36

10 1 10 45 120 210 252 210

11 1 11 55 165 330 46212 1 12 66 220 495

The columntotals are 1,1,2,3,5,8,13,21,34,55,89,1445.... These numbers lsoseemto appear inthenext hree xamples,s wellas inExamples70 and 80.

Example41 The ceiling f, east nteger ot essthan, n-1)/2, for

n= 0 1 2 3 4 5 6 7 8 9...

is 1 1 2 3 5 8 13 21 34 55....

Example 42 If a(n) is the sumof the divisors f n, thena(n)/n measures heabundancy f n. Everynumber with bundancy> j musthaveat leastk factors,where, or

2 3 4 5 6 7 8 9...

k= 2 3 4 6 9 14 22 35....

The differencesfthis astsequence re 1,1,2,3,5,8,13,....





VOL. 63, NO. 1, FEBRUARY1990 7

Example 47 The number f rootedtreeswith n vertices,ust one of whichislabelled.

L~~~~~~~

L L L

tLL LLL

1 ~~~~~~25

_ 0t_ I _ L L

13

Example48 The number fdisconnectedraphswithn + 1vertices.

@ * @@*S *AL* 0 0 0 *-O0 0 6

11

0/

1 2 5

* *A S,_TKA, A%x'\

13

Example49 The number fconnected raphs nn + 2 vertices ithustonecycle.

1 2 5

*13

For manyotherexamples nvolving raphs, ee [22],whichdoes not, however,includeExamples47-49.

Example50 The coefficientsnthepower eries olution

x2 x3 2X4 5 13x6y= 1+ x2+ ! + 5ix + 6! +

ofthedifferentialquationD2y = exy.



8 MATHEMATICS MAGAZINE

Example 51 The sequence n = an1 + nan2 (n 1) with 1 =ao= 1/2

1 1al 2 +1 2- = 1

a2= 1 +2x 2 = 2

a3=2+3X 1=5

a4=5+4X2= 13.

Example 52 The sequence bn (n - 1)2n-2 + 1, n > 1.

bl=0x2-'+ I = 1b2= 1 x20+ 1 =2

b3=2x2'+ 1=5

b4=3 x22+ 1= 13.

Example 53 How manydistinct ums, f(n), maytherebe of n differentrdinalnumbers?Obviously,f(l) = 1. However, f(2) = 2, because ordinal ddition s not

commutative. orexample, + X = X 0 X + 1.Youmight uessthatf(3) could be aslarge as 3! = 6, but in factyou can't have morethan5 distinct ums of 3 differentordinals. he answers

forn= 1 2 3 4 5 6 7 8...

aref(n) = 1 2 5 13 33 81 193 449...

perhaps he samesequenceas Example52. Or perhapsnot.

Example 54 The values of the polynomial n2 231n+ 1523 forn= 0,1,2, .. are1523, 1301, 1097, 911, 743, 593, 461, 347, 251, 173, 113, 71, 47, 41, 53, 83, 131,197,.... Try also the polynomial7n2 1701n+ 10181.

Example55 Whatare the next hree ermsnthesequence

(1),2,3,5,7, 11,13,17,19,23,29,31,37,41,43,47,53,...?

Example56 The integer artofthe nthpowerof3/2

n 0 1 2 3 4 5 6

(3/2)n 1 1.5 2.25 3.375 5.0625 7.59375 11.390625

0 1 2 3 5 7 11




Example57 The number ftreeswithn edges, nd height t most .

1 1 2 3

5 7

11

Example58 The number fpartitionsfn

n= 0 1 2 3 4 5 6 7 8 9...p(n)= 1 1 2 3 5 7 11 15 22 30....

Example59 Ifweformuccessive ifferencesfthepartitionunction:

1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 6270 1 1 2 2 4 4 7 8 12 14 21 24 34 41 55 66 88 105 137 ...1 0 1 0 2 0 3 1 4 3 7 3 10 7 14 11 22 17 32 ...-1 1 -1 2 -2 3 -2 3 -2 5 -4 7 -3 7 -3 11 -5 15 ...

we see thatthethird-orderifferenceslternaten sign.

Example 60 If you expandthe product 1 - x)(1 - x2)(1 - x3)(1 - X4)*, you

get,successively

1-x1 - xx2 + X3

1--x2?+X4+X5 _x6

1-x-x2 + 2x5 x8 -x9 + x10

and a coefficient has appeared. Indeed, at stage 10, a coefficient appears.However, urtheralculationppears o cancel theseout, eaving

1-X-X2 +X5+x7-xl2-xl5+

Are there nycoefficientsther han0, ? 1 in thefinal esult?




Example 61 For each integer xponent, , is there n integerm > 1 such that hesum ofthe decimaldigits fm' is equal to m?

21,92 = 81, 83 = 512,74 = 2401,28 = 17210368, 86 = 34012224,187 = 61222032,

468, 549, 8210, 81",10812,013, 9114,10715,13316 8017,17218, 019, 020, 9021.

Example 62 A Nivennumber as beendefineds one which sdivisible y the sumof tsdecimal digits, uchas 21 and 133. Is n! always Nivennumber?

4! = 24, 5! = 120,6! = 720,7! = 5040,8! = 40320,9! = 362880,10!= 3628800.

Example 63 Can youchoose sequenceof real numbers rom he nterval0, 1) so

thatthefirst wo ie indifferentalves, hefirsthreendifferenthirds,he first ourin differentuarters,ndso on? For example,

0.71, 0.09, 0.42, 0.85,0.27, 0.54,0.925, 0.17,0.62,0.355, 0.78,0.03,0.48,....

Ifyou run ntodifficulty,ouareallowed o adjust arliermembers fthesequence, fyoulike.

Example 64 Surely very ddnumber greaterhan 1, ifyou don'twant to count 1

as a prime) s expressibles a prime lus twice square?3 + 2 02 3 + 2 12,5 + 2 12,7 + 2 12,3 + 2 22, 11 + 2 12,7+ 2 22, 17+ 2 02, 11 + 2 22,3 + 2 32, 5 + 2 32, 23 + 2- 12...

Indeed, somenumbers,uchas 61, haveseveral uchrepresentations.

Example 65 Is n! always xpressibles the differenceftwo powers f 2?

0! = 1! = 2' - 2? 2! = 22 2', 3! = 2 - 2', 4! = 2 - 23, 5! = 2 - 23

Example 66 It'swellknown hat ! = 52 - 1, 5! = 112-1 and 7! = 712-1, butnotso wellknown hat fyoutakethe next quarebigger hann! thedifferences alwaysa square:

6! = 272 - 32, 8! = 2012 - 92, 9! = 6032 -272,

10! = 19052- 152, 11! = 63182- 182I...

Example 67 The valuesofsin2(k7/12), ork= 0, 1,... 6 are

k= 0 1 2 3 4 5 6

sin2k7/12) = 0 (2-F3)/4 1/4 1/2 3/4 (2 + 3)/4 1

It's alsowellknown hat

J/ sin2nxdx= (2n- 1)(2n-23) ..3 1



VOL. 63, NO. I, FEBRUARY 1990 11

If youcalculatethe ntegral y thetrapezoidal ule,using6 equal subintervals,ouwill gettheanswer

~22?-1? (2? )? (2 3)n + 3n+ 2n + 1}7/12 4 ,

which s exactforn= 1, 2, 3, 4,5,6 and 7.

Example68 The continued raction orq 2leY s

7T2 1 1 1 1 1 1 1

e 5? 1+ 1? 5? 1? 1? 5? 1? 1?

Example 69 Define a sequence by P(1) = P(2) = 1, and for n > 2, P(n) =

P(P(n - 1)) + P(n - P(n - 1)). Thefirst2 termsre 1,1, 2,2,3,4,4, 4, 5, 6, 7,7,8,

8, 8, 8, 9, 10, 11,12,12,13, 14,14,15, 15,15, 16, 16,16, 16, 16. Note that P(2) = 1,P(4) = 2, P(8) = 4, P(16) = 8, and P(32) = 16.

Example 70 A similar equencestartswithQ(1) = Q(2) = Q(3) = 1, and the samerecurrenceforn > 3, Q(n) = Q(Q(n - 1)) + Q(n - Q(n - 1)). The first 4 termsare1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5,5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 11,11, 12, 12,12, 13,13,13, 13, 13, 13. Notice that Q(2) = 1, Q(3) = 1, Q(5) = 2, Q(8) = 3, Q(13) = 5,Q(21) = 8 and Q(34) = 13.

Examples40-52 and70 perhaps ontainmanifestationsfthe Fibonaccinumbers.Almost s ubiquitous re the Catalannumbers,2n)!/n!(n + 1)!,

1, 1,2, 5, 14,42, 132,429, 1430,48625....

How manyof Examples 1 to79 are genuine?

Example71 The number fmountain angesyoucan drawwithn upstrokesnd ndownstrokes:

1 1 ~~~~~~~~~~~~~~~~2

-- AA

14



1 2 MATHEMATICS MAGAZINE

Example 72 The number fwaysofmakingn folds n a stripof n + 1 postagestamps,wherewe don'tdistinguishetweenfrontnd back, topand bottom, rleftand right:

0 01 012 021 0123 0132 0231 0321 10321 1 2 5

01243 01432 02431 03214 03421 04123 04321 10243 1034210432 1403201234 01342 02341

14

Example73 The number fdifferentroups, p to isomorphism,forder n is,

forn= 0 1 2 3 4...no. ofgroups 1 1 2 5 14....

Example 74 The number fways 2n people at a round able can shakehands npairswithout heir ands rossing.

1 1 2 5

14

Example75 The number ftriangulationsfthe pherewithn + 4 points.

2

5

We leave the readerto verifyhat there re just 14 distinct riangulationsf thespherewith8 points.




Example76 The number frooted lanetreeswithn edges."Plane" means hatwedistinguishetween eft ndright.

2 5

14

Example77 Thenumber fwaysofdistributingdifferentbjects nindistinguish-ableboxes,with tmost hree bjectsna box.

LiS L ~LJIkLLLL1 2~~~~~~~~11

D

I l ~~~~~~~~~2 a

14

Example78 The probability,orn= 0, 1,2,. ., that function (k), withdomaink = 0,1 ..., n and range [0,1], is convex,is

1 1 2 5 14

(0!)2 (1!)2' (2!)2 (31)2' (4!)2 2**

Example 79 The incomplete Bessel function of order one has power series expansion

l(2x )/X 3 I + X 2X2 5X6 14x8

2! 4! 6! ? 8

Example 80 Examples 37 to 40 involved Pascal's triangle, whose entries are the

binomial coefficients. We can use a similar array to expand (1 + x + x2),, giving

trinomial coefficients. Each entry is the sum of the three nearest in the previous row.

1 1 11 2 3 2 1

1 3 6 7 6 3 11 4 10 16 19 16 10 4 1

1 5 15 30 45 51 45 30 15 5 1

1 6 21 50 90 126 141 126 90 50 21 6 1

1 7 28 77 161 266 357 393 357 266 161 77 28 7 1

1 8 36 112 266 504 784 1016 1107 1016 784 504 266 112 36 8 1

19 45 156 414 882 1554 2304 2907 3139 2907 2304 1554 882 414 156 45 9 1




The central rinomialoefficient,n,

1,1,3,7, 19,51, 141,393,1107,3139,...

almost reblesn size at each step: fwe calculate3an an+ we get

2,0,2,2,6,12,30,72,182,...

which are pronicnumbers,m(m + 1), form= 1,0,1,1,2,3,5,8,13,....

Answers

36. This is the polynomialn 1) + (n 1) + (n 1 + (n 1) + (n ) of Example 5 of[21], and represents he number fpieces you can cut a circular ake into byslicing etween very airofpoints hosenfrom around hecircumference.t

is also the number fregions hat4-dimensionalpace is chopped ntoby n - 1hyperplanesn general osition. he sequence s #427 n [45]: 1, 2, 4, 8, 16, 31,57, 99, 163, 256, 386, 562, 794, 1093, 1471,....

37. This verybeautiful etting orExample1 of [21] was observed 0 years go byWilliam Watkins, ow co-editor f Coll. Math. J.Gauss has told us that thenumber f sides n a regular olygonwhich an be constructed ith traightedgeandcompass s of hape2m17JFn,here heFn redistinctermat rimes2n+ 1.Only five uch,0 < n < 4, areknown ndsome people believethatno otherswilleverbe found. o thepattern reaks own t row32. Fermat hought hat 32+ 1

was prime, ut Euler discovered he factorization41X6700417.38. This is the Mann-Shanksrimalityest 36]. Surprising,fnotpractical. an youprove t?

39. This s an observationfGerryMyerson: hat hebold numbers re the compositenumbers. owever, hisbreaks own n row13, because 11)= 211. 13 and

(l )= 223. 11*13arenot quarefree.40. This well-known elation etween Pascal's triangle nd Fibonacci numbers s

easily een topersist,ince achentrysthe umoftheentriesnthepreviouswocolumns ftheprevious ow, o each total s the sumof the twoprevious otals.

41. This is adapted froman inequalityof Larry Hoehn, of ClarksvilleTN.The coincidence s quite surprising,ince re 1.64872 and the goldenratio(1 + V5)/2 = 1.61803arenotremarkablylose. For n= 10,11,12,... the terms91,149,245,... beginto diverge romheFibonacci equence89,144,233,....

42. In [32],RichardLaatschshowsthat thesequencecontinues 5,89,142,230,....withdifferences

20 30 53 88 143236387641 10611763 273749038202 13750 23095...

which tayclose to the Fibonaccinumbers

2134 55 89 144233 377 610 987 159725844181676510946 17711 ..

for while,buteventuallyendto infinity orerapidly.43. See sequence X912 in [45]. This is stilla notorious pen question:thereare

extensive ables [30, 34, 49, 50]. Duringrevision f thisarticle,Dick Lehmerkindly an a programn a 75 Vax, ndfoundnocounterexampleithp lessthana million.

44. The sequencecontinues 7,26,34,45,54,67,... and is denser han he Fibonaccisequence. t is #254 n [45],butthe referencehere smisleading.he sequence

doesn'tsolve Amer.Math.Monthly roblem 1910 [1966, 775; partial olution




1968, 80-81] because the differencesre not unique: e.g., 17- 8 = 26 - 17=54 - 45. Nor s itthe uxiliaryequence{r,2= {4, 5, 9, 10, 11, 16, 18, 22, 23, 24,25, 27, 28, 29,... }, used to construct he Sierpinskiequence,#425 in [45].There's another pen question ere: find he smallest ossible symptotic rowth

for sequenceof ntegersuch thateverypositive nteger ccurs uniquely s adifference.

45. This also fails o continuewith he Fibonacci equence.The numbers f arrange-mentswith7,8,9,... pennies re12,18,26, ... Thesearrangementsere tudiedbyAuluck 2]; see sequence#253 n [45],and compareExample34 in [21].

46. These are indeed theodd-rankingibonaccinumbers, 2 -1' sequence#569 n[45], whichhave the property

U2n-1= U2n-3+ 2U2n-5 + 3U2n-7+ + (n - ul+1

which can be seen to be the number f ways thata row of n penniesmay besurmounted y an arrangementithn- k in its bottom ow, n any one of kpossiblepositions, herek= 1,2,. ., n - 1 or t's notsurmountedt all (k = n).

47. These are not the alternate ibonaccinumbers, .g., the numbers f such treeswith 5,6,7,... vertices re 35,95,262,... See sequence#570in[45] or p. 134in [43].

48. Norare these.The nextfewmembers fthesequenceare44, 191, 1229, 13588,288597,.... See sequence#574 n [45],or [24].

49. Neither s this thesequenceof alternate ibonacci numbers, ut continues 3

(one short!), 9 (correct!),40 (7 toomany), 57, 1806,5026,.... See sequence#568 n [45] or page 150in [43].50. Nor is this, which continues 6,109,359,1266,4731,18657,77464,...; see se-

quence#572 n [45]and Tauber'spaper 48].51. Nor again, since a5 = 13+ 5 x 5 = 38, a6 = 116, a7 = 382,...; see sequence

#573 n [45].52. Neither rethese,b5= 4 23+ 1= 33, b6= 5 24+ 1 = 81, b7= 6 25+1 = 193,

b8= 7 26+ 1 = 449, alternate ibonacci numbers, ut theydo featureforawhile) n the nextExample:

53. which gotfrom ohn onway. f g(k) = k*2k-I + 1, then

f(n)= max f(n-k)g(k),O<k<n

and, for n < 8, f(n) is indeedequal to g(n - 1). Thereafterhe situation etsmorecomplicated, uta simple uleeventuallymerges: orn = 9,10,11,12,13,f(n) = 332,33 81,812,81. 193,1932, nd,forn > 14, f(n) = 81ff(n 5), exceptthatf(19) = 1933.

54. Severalreaders f[21] said that shouldhave includedEuler'sfamous ormula,n2+ n + 41,which ivesprimes or < n < 39,notnoticinghatExample 1 was

just that,exceptforthedisguise fomittinghe tell-tale 1 (n = 0). For someastonishing xamples f The Strong aw in thisconnection,ee thepapersofStark 46, 47]. Thepresent olynomials a slight daptation f onedue toSidneyKravitz, nd is found y replacing inEuler'sformulay38 - 3n. Surprisingly,this tillgivesprimes or < n < 39, althoughhirteenf them re notamong heoriginal orty; = 40 and41 give6683= 41 X 163 and 7181= 43 X 167.

The polynomial 7n2 1701n+ 10181 was discoveredrecently y GilbertFung. If youworkmodulop forprimes < p < 43, you'llfind hat t's neverdivisibleby suchprimes. t takesprimevaluesfor0 < n < 42, beatingEuler'srecordby two.Noticethat he discriminantf Euler'spolynomials - 163,and




thatof Kravitz s - 32 X 163,whileFung'spolynomial as to have a positivediscriminant,79373.

55. Such questions rehardly air, inceargumentsan be advancedfor ontinuingsequences n anywayyouwish.Some answers re moreplausible hanothers,

however,nd theonethatPersiDiaconishopedyouwouldmiss s 59,60,61,...,theorders fthesimplegroups!56. Another utile ttempt o foolyou ntothinkingftheprimes. he nextmember

is 17, then25,38,...; see sequence#245 n [45].57. This is not thesamesequence s thepreviousxample, ut see thenext!58. To see thecorrespondenceetween his nd theprevious xample, otethat he

number f'verticestheight ne isthenumber fparts,nd thevalences ftheseverticesre the sizesoftheparts. equence#244 n [45]; seealsopage 122 n [43]and page 836 in [1].

59. This examplewas sentbyGerryMyerson.t canbe proved hat hedifferencesfanyorder repositive rom omepoint n,but thatpoint ecedes ather apidlysyou take higher rderdifferences.he nextfewthirddifferencesre - 4,17,-2,24, -4, 32,1,38,5, .. and are positive rom owon. The fourth ifferencesalternaten signuntil he67th, fterwhich hey repositive.

60. This is Euler'sfamous entagonal umbersheorem:

00 00

J7 1--

n) = L (_ I)kXk(3k-l)/2n=1 k=-oo

See theorem 53 in [26], for xample.61. NormanMegillofWaltham,MA,finds uchm foreach n < 104. For n = 105,

however, o suchm exists.62. ThisquestionwasaskedbySamYates.CarlPomeranceuggested hat ounterex-

amplesmightbe expectedby the timen has reached500, and indeedYatesfound hat432! is nota Nivennumber,ince hesumof tsdigitss32 X 433,and433 is prime.

63. The givensequencecan be continued, .97,0.22,0.66,0.32, ut BerlekampndGraham 3]have shown hatnosuch equence xistswithmore han17members!

64. This special case of the Hardy-Littlewoodroblemwas mentionedby RonRuemmler fEdison,NJ,whobelieves hat hefirstxceptions 5777,andasks fit is also the last! It is knownfrom he workofHooley[27], Miech [37], andPolyakov42] that hedensityfexceptionss zero.

65. IgnaceKolodner ot this rom aroldN. Shapiro n an NYU Problem eminar n1949. It's left o thereader oprove hatn! is never gainthedifferenceftwopowersoftwo.

66. This was observedby LarryHoehn of Clarksville, N. It fails for12! but13!= 789122 2882, 14! = 2952602 4202, 15! = 11435362 4642, 16! =

45741442 18562.t's doubtfulfthisoften ccursfrom ereon (notethatyoumust ake the next quarebigger hann!),but tmaybe hardtoprove nything.

67. This s alsocorrect orn= 8,9, 10,and11,but forn= 12 weget 1352079X)/224insteadof 1352078v)/224,outby3 parts n fourmilion!The trapezoidal ulegivestheright nswer fyouuse k subintervals,rovided n is lessthan4k: see[28],for xample.DavidBloom uggestedhat fourmillion"hotldread "sixteenmillion": intended he relative rror, 2.958/4000000: the actual error s- 2.996/16000000:more xamples ftheStrong aw!

68. If thispattern, oticedbyJamesConlan [8],wereto continue,we wouldhave(5 + 37 )e- = 27T2.Close,but no cigar!



VOL. 63, NO. 1, FEBRUARY1990 17

69. The sequencethathitthe national resses n both idesof theAtlantic, .g. [6],publicizingthe Conway-Mallowsncounter. have an earliermanuscript fConway in which he has writtenin anothernotation) P(2k) = 2k- 1 (easy),P(2n) < 2P(n) (hard), P(n)/n -2

' (harder)."twas theproof fa precise orm

of this ast statement hat lmostwonMallows venmoremoney hanConwayintended. apersmentioninghis equence nclude 16, 35].

70. Yes, theFibonaccipatternontinues40]. David Newman howed his o DavidBloomas a conjecturen 1986.

Nine of the final enexamples re intended o look ike theCatalannumbers;sequence#577 n [45]. Atfirstt is a matter fsomesurprise hat

I= 12n)

is alwaysan integer. n connectionwith ome recent orrespondence41], JohnConwaymakes he moregeneral bservationhat

(m, n)(m + n - 1)!m!n!

is an integer,where m, n) is the g.c.d.ofm and n, because

in(m?n-l)! (+n I and n(mn+n-1)! =) n-Im!n! m- I1

n m!n! n - 1

are both ntegers. his alsoanswers questionnB33 of[20], whereNeil Sloanegavetheexamplen= 4m+ 3.

Catalan numbers ccurin manywidelydifferentooking ontexts: ee [18],withnearly 00 references,nd [31], with listof31 structures,othobtainablefromH. W. Gould, Department f Mathematics,West VirginiaUniversity,Morgantown,WV, 26506. An articlewith a good bibliographys [5]. Several" proofswithoutwords," howingheequivalence fseveral f thestructures,illappear in [9].

71. This is a genuine xample f the Catalannumbers. he mountain anges re the

same as pathsfrom0,0) to (n, n) whichdo not crossy= x, or incoming iedballots n which necandidate s never ehind, rsequences f zeros ndones,orof + Is, subjectto appropriate um conditions, .g., randomone-dimensionalwalks nwhichyou nevergo tothe eft f theorigin; ee [13].

72. Thissequence, #576 n [45], s not, nd continues 9 (not38, as stated n [14]),120,358, 1176,3527, 11622,36627,121622, 89560,..., see [29].

73. The numbers fgroups forders 5 and 26 are51 and 267 [23]. Thissequence,#581 n [45],continues 328 [51],56092[52].

74. This is genuineCatalan again: see [39].75. But this ne, sequence#580 n [45],has beencalculated or nlyfourmore erms

[4, 12, 19]. Ofthe

1,1,2,5,14,50,233,1249,7595 triangulations,

only0,0,1,1,2,5,12,34,130 containnovertex fvalence3.76. is a genuiinemanifestationf theCatalan numbers7, 25],but77. is not:sequence#579 n [45]continues6, 166,652, 2780,12644,61136,312676,

1680592,... [38].78. The probabilityorgeneraln is indeedcn/(n!)2 10].79. In [10] we askedwhat was theexponential enerating unction or heCatalan




numbers. ouis W. Shapiro bserves hatoo X2n

Cn (2n)! =11(2x)/x

where is the modified essel functionforder ne: seeformula .6.10.on page375 of[1]. In thepaper [44]he obtains esults or atticepathswhich taybelowgiven points, rrangedwith ncreasing bscissas nd ordinates,omewhat nalo-gous to theconvexfunctionsf 10].

Beforewe say goodbye o the Catalannumbers, ere's an observation hichmaynot be widelyknown. t originatedn a discussionwithJohnConway onlysixmonths go. What is well known s that he Catalan numbers re associatedwithparenthesization.y thatmostpeople mean the numbers fpossible rdersof n nonassociative perations, suallyndicated y n - 1 pairsofparentheses:

n=0 a n=l ab n=2 (ab)c or a(bc)n = 3 ((ab)c)d (a(bc))d a((bc)d) a(b(cd)) (ab)(cd)n = 4 (((ab)c)d)e ((a(bc))d)e (a((bc)d))e (a(b(cd)))e ((ab)(cd))e

((ab)c)(de) (a(bc))(de) (ab)((cd)e) (ab)(c(de)) a(((bc)d)e)a((b(cd))e) a((bc)(de)) a(b((cd)e)) a(b(c(de)))

and so on. But they are also the numbers f ways of arrangingn pairs ofparenthesess a pattern,ustfor heir wn sake:

n=O n=1 () n=2 (() or ()n( )0 (())) (() ( 00(0)(3)())) 0 ( )

n = (()) ( )() ()( )) (() ) ) ) ( ( )( ))( ))

( )()()())(( ( )) )( ) ( )( )) ) ( )()( )) (( )() )) ( )() )

An examinationf the symmetriesn thetwo cases makes t unlikelyhatyou'llfind directcombinatorialomparison. ne-one correspondencesetween theformermanifestationnd otherCatalanmanifestationsre well known. he latterare easily een to be incorrespondence ith hepairsof people shaking ands nExample 73, and with he mountainsn Example70.

80. Jack Good [17] has given an asymptotic ormula or the centraltrinomialcoefficient:

an -3 1;1-3:1 2 + 0(n-3)2~7Tn 5122

which howsthat he eft ide of the"identity"

e 3an-an+?1=un1(un?1) ?

growsikec x 3nX n-3/2, whereas heright ide grows ike r2n/5,whereT is thegolden ratio, T2= (3 + F5 /2. Furthercalculation shows that a10= 8953,3a9 - a10 = 464, while u8(u8 + 1) = 21 x 22 = 462. The asymptotic formula isgoodto thenearestnteger or uite argevaluesof n.

This example was sent by Donald Knuth.Euler [11] was one of the earlierdiscoverers f The Strong aw of SmallNumbers, nd called this

exemplum emwrabilenductionisallacis.

On the samepage he gives he Fibonacciformula hat's ften ttributedo Binet.




Coda I showed this exampleto George Andrewsduringthe recentBatemanRetirement onferencet Allertonark, llinois.Half-an-houraterhe cameback withwhatEulerreally houldhave said. He defines hetrinomialoefficientsentrallyy

(1x?+x2) E ( n

and proves hat, fFnis thenthFibonaccinumber,hen

Fn +1)=OX (1 + 1 )2 ( OA+ 2 2)

For - 1 < n < 7, theonlynonzero erm n therights X= 0, whichaccountsforEuler'sobservation,ince

3(0)2_2 n )=2( 0 -2( 1

Andrewswillpublish heq-analog fthis heoremhortly.

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Artigo de 1990 de Richard Guy Sobre a Lei Forte Dos Pequenos Numeros

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