Generalized expansions of real numbers
by Simon Plouffe

april 2006

 

 

I present here a collection of algorithms that permits the expansion into a finite series or sequence from a real number x(0)  R, the precision used is 64 decimal digits. The collection of mathematical constants was taken from my own collection and theses sources [1]-[6][9][10]. The first and simplest algorithm is the continued fraction algorithm, see Wikipedia for an introduction to continued fractions.

Most of the algorithms will produce a sequence of integers when x  R and can be written as a 2 terms recurrence. If x(0) is the initial value then y(n) will be the terms of the sequence. If x is rational the sequence is finite and if x is irrational the the sequence is infinite but with 64 decimal digits enough terms are computed for detecting simple patterns as with some quadratic irrational like √2 or . Other numbers like exp(1) : 2.71828… do have a pattern which is easily recognizable but most real numbers do not. The goal of this computation of sequences from real numbers using different algorithms is to discover or find if there could be any patterns at all with other algorithms. 

The natural question that comes to mind is : is there any closed formula or generating function for those sequences?. For this I can use GFUN [8] which is a Maple package or with Mathematica as well to answer the question [7]. A known example is sinh(1) = 1.175201193643801456… which is : 1, 6, 20, 42, 72, 110, 156, 210, 272, 342, … when expanded into the Engel expansion. That sequence appears to be the coefficients of the series expansion of this rational polynomial :

 

So the coefficients are given by the polynomial : 4n2+2n, n > 1 by using Montmort formula.

Unfortunately this is a lucky example because for tanh(1) = 0.761… with the Engel expansion we  have : 2, 2, 22, 50, 70, 29091, 49606, 174594, 260086, … which does not coerrespond to any known closed formula where the continued fraction expansion of that same numbers is : 0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ….

 

 

 

 

The main algorithms

 

Where [x] denotes the integer part function and {x} the fractional part, y(n) are renamed a(n).

To see each equation bigger just click over with the mouse.

 

   which gives  

 

 which gives :  

 

 which gives :  

 

 

which gives :  

 

 

 

      which gives :  

 

 which gives :  

 

 which gives :  

 

 which gives :  

 

 

Here is a list of 188 mathematical constants and their expansions. The naming convention used here reflects the fact that these constants are now included in my new database of 1.099 billion constants and not yet implemented in my Inverter on the web [9].


1.002008392826082214417852769232412060485605851394888756548596615e0 Zeta(9)
1.008349277381922826839797549849796759599863560565238706417283136e0 Zeta(7)
1.009559712542709408179200409989251636051890411928097814194168320e1 Psi(1,1/3)
1.025836496752890287411804996499918568734974052810169947019280777e0 GAMMA(23/24)
1.034653881897437911619794298464638254670307984344385254503070281e0 MertensB2
1.036927755143369926331365486457034168057080919501912811974192678e0 Zeta(5)
1.055546564813466302313701484734262209810379114858428728414892620e0 GAMMA(11/12)
1.057250875375728514571842354895877959024053937569807899612103522e0 Shi(1)
1.076539192264845766153234450909471905879765632901150866985681470e-1 OneNinth
1.084101512231113615112908114064150911221580909390909100000000000e-2 Trott
1.098612288668109691395245236922525704647490557822749451734694334e0 log(3)
1.128787029908125961260901090258842013326787441664755451752083514e0 GAMMA(5/6)
1.149942818607399066388560985243920097987661520136529721953851784e1 Pi*csc(Pi/12)/GAMMA(11/12)
1.173724105211061175126493432503988576654303066406559411204845301e0 GAMMA(19/24)
1.176280818259917506544070338474035050693415806564695259830106347e0 Salem
1.186569110415625452821722975947237120568356536472054335954254299e0 KhinchinLevy
1.202056903159594285399738161511449990764986292340498881792271555e0 Zeta(3)
1.225416702465177645129098303362890526851239248108070611230118938e0 GAMMA(3/4)
1.226742010720353244417630230455361655871409690440250419643297301e0 FibonacciFactorial
1.234567891011121314151617181920212223242526272829303132333435364e-1 Champernowne
1.259921049894873164767210607278228350570251464701507980081975112e0 2^(1/3)
1.266065877752008335598244625214717537607670311354962206808135331e0 BesselI(0,1)
1.282427129100622636875342568869791727767688927325001192063740022e0 GlaisherKinkelin
1.28435556102175533436225346195190183345531497710084581171260000e-2 ZetaR(4)
1.285050699666050517584462428785779493488009654041863700774691745e0 GAMMA(17/24)
1.303577269034296391257099112152551890730702504659404875754861391e0 Conway
1.306377883863080690468614492602605712916784585156713644368053760e0 Mills
1.316957896924816708625046347307968444026981971467516479768472257e0 log(2+3^(1/2))
1.317641154853178109817352322513585951250734323295251678792547422e-3 HeathBrownMoroz
1.324717957244746025960908854478097340734404056901733364534015050e0 PisotVijayaraghavan
1.332582275733220881765828776071027748838459489042422661787130900e0 MertensB3
1.339784153574347246599152586514886052775242249788182806663015068e0 Totient
1.354117939426400416945288028154513785519327266056793698394022468e0 GAMMA(2/3)
1.355461804706408743863441535910970976036790271047993178161793668e0 FeigenbaumKappa
1.374802227439358631782821879209657256986307759467366665441760509e0 exp(1/Pi)
1.395485972479302735229500663566888068954103728144661190817472156e0 HardHexagonsEntropy
1.403936827882178320576206074137209354537638761521890067378557540e0 OrthogonalArrays
1.413472514173469379045725198356247027078425711569924317568556746e1 Riemann1stZero
1.414213562373095048801688724209698078569671875376948073176679738e0 2^(1/2)
1.442249570307408382321638310780109588391869253499350577546416195e0 3^(1/3)
1.444667861009766133658339108596430223058595453242253165820522664e0 exp(1/exp(1))
1.451369234883381050283968485892027449493032283648015863093004558e-1 Soldner
1.456074948582689671399595351116543557653178374847131540270702437e0 Backhouse
1.460354508809586812889499152515298012467229331012581490542886088e0 Zeta(1/2)
1.461632144968362341262659542325721328468196204006446351295988409e0 MinimumGamma
1.467078079433975472897798484707229953449903322414887777399685818e0 Porter
1.475836176504332741754010762247405259511345238869178945999223128e-1 arctan(1/2)/Pi
1.484336564670078282258650774907113718875558417448068894425070000e-1 ZetaR(2)
1.515426224147926418976043027262991190552854853685613976914074640e1 exp(1)^exp(1)
1.528709197087111005047302663828105693597737976624121924443565292e0 GAMMA(7/12)
1.584962500721156181453738943947816508759814407692481060455752654e0 log(3)/log(2)
1.590636854637329063382254424999666247954478159495536647132287985e0 BesselI(1,2)
1.606695152415291763783301523190924580480579671505756435778079554e0 ErdosBorwein
1.609437912434100374600759333226187639525601354268517721912647891e0 log(5)
1.618033988749894848204586834365638117720309179805762862135448622e0 Golden Ratio or 1/2+sqrt(5)/2
1.639914804874884209492639941554099128143343636389220916737899189e0 GAMMA(13/24)
1.64958415402929159899676131363885182748790994383473214781100000e-3 ZetaQ(4)
1.652891650281069480098240646585387817777383221122454651371715916e0 (1+3^(1/2))^(1/2)
1.654211437004509292139196602427806427640363803352017836665223064e-1 Zeta(1,-1)
1.705211140105367764288551453434508160762027651653469099994284907e0 Niven
1.732050807568877293527446341505872366942805253810380628055806979e0 3^(1/2)
1.745405662407346863494596309683661067294936618777984256595013774e0 KhinchinHarmonic
1.747564594633182190636212035544397403485161436624741758152825351e0 MadelungNaCl
1.74762639299443536423113314665706700975412121926149289888672000e-1 ZetaP(3)
1.772453850905516027298167483341145182797549456122387128213807790e0 Pi^(1/2)
1.781072417990197985236504103107179549169645214303430205357665877e0 exp(gamma)
1.782213978191369111774413452972549340791731909773239381024959956e0 Grothiendeck
1.787231650182965933013274890337008385337931402961810997784781471e0 KomornikLoreti
1.822825249678847032995328716261464949475693118894850218393815613e0 MasserGramainDelta
1.831258984937131485342173397873781627441298090376237965477411646e0 FeigenbaumC
1.839286755214161132551852564653286600424178746097592246778758639e0 Tribonacci
1.851937051982466170361053370157991363345809728981154909804783782e0 Si(Pi)
1.895117816355936755466520934331634269017060581732707591646228432e0 Ei(1)
1.927561975482925304261905861736622168698554255163384727146647038e0 Tetranacci
1.932235335236375341775586632622696654743936114291191986048917058e0 Pi*csc(11/24*Pi)/GAMMA(13/24)
1.945280494653251136152137302875039065901577852759236620435639113e-1 DuboisRaymond
1.962858173209645782868795128675183526649593017162219421130715240e0 ReciprocalLucas
2.029883212819307250042405108549040571883378615060599584034978214e0 Figure8HypebolicComplement
2.053834420303345866160046542753384285715804445410618245481483337e-3 gamma(3)
2.078795763507619085469556198349787700338778416317696080751358831e-1 exp(-1/2*Pi)
2.078862249773545660173067253970493022262685312876725376101135571e-1 -Zeta(-1/2)
2.102203963877155499262847959389690277733434052490278175462952040e1 Riemann2ndZero
2.127557058602221971572278439344293350609569221813237534859862516e0 Pi*csc(5/12*Pi)/GAMMA(7/12)
2.193839343955202736771637754601216490310472934069082075779786131e-1 Ei(1,1)
2.236067977499789696409173668731276235440618359611525724270897245e0 5^(1/2)
2.245915771836104547342715220454373502758931513399669224920300255e1 Pi^exp(1)
2.279585302336067267437204440811533353285841102785459054070839752e0 BesselI(0,2)
2.294856591673313794183515831344311288713163799441668673275814030e0 product(Zeta(n),n=2..infinity)
2.314069263277926900572908636794854738026610624260021199344504641e1 exp(Pi)
2.346248769318331988138571146958629493043336513400461016473979848e1 Pi*csc(1/24*Pi)/GAMMA(23/24)
2.357111317192329313741434753596167717379838997101103107109113127e-1 CopelandErdos
2.501085758014568876321379099256282181865954967255799667249654201e1 Riemann3rdZero
2.502907875095892822283902873218215786381271376727149977336192057e0 FeigenbaumAlpha
2.584981759579253217065893587383171160088051651852630917321544988e0 Sierpinski
2.588190451025207623488988376240483283490689013199305138140032073e-1 sin(1/12*Pi)
2.614972128476427837554268386086958590515666482611992061920642139e-1 MertensB1
2.665144142690225188650297249873139848274211313714659492835979593e0 2^(2^(1/2))
2.678938534707747633655692940974677644128689377957301100950428328e0 2/3*Pi*3^(1/2)/GAMMA(2/3)
2.683150900474071801449936758645148251038610920554435493093680868e0 FeigenbaumD
2.685452001065306445309714835481795693820382293994462953051152346e0 Khinchin
2.718281828459045235360287471352662497757247093699959574966967627e0 exp(1) ou E
2.739441957392716171714591527242859192737251877298819862919173838e-2 Trott2nd
2.807770242028519365221501186557772932308085920930198291220054810e0 FransenRobinson
2.955765285651994974714817524123194588375492304663596595350472479e0 Otter
3.010299956639811952137388947244930267681898814621085413104274611e-1 log(2)/log(10)
3.036630028987326585974481219015562331108773522536578951882454815e-1 GaussKuzminWirsing
3.074948787583270931233544860710768530221785199506639282983083963e-1 HardyLittlewoodC4
3.081505560003435202605938614633736771064504215893189288306692702e0 Pi*csc(7/24*Pi)/GAMMA(17/24)
3.141592653589793238462643383279502884197169399375105820974944592e0 Pi
3.217505543966421934014046143586613190207552955576561914328030594e-1 arctan(1/3)
3.226340989392446705795316925482370665709505796658327099618112524e-1 FellerTornier
3.359885666243177553172011302918927179688905133731968486495553815e0 ReciprocalFibonacci
3.405373295509991428262731844329028967106082171243020977632361054e-1 PolyaRandomWalk3D
3.407069165627256142219458262827180653554034438015032116191033828e0 Magata
3.569945671870944901842005151386498936763836911514832378107975530e0 FeigenbaumMu
3.625609908221908311930685155867672002995167682880065467433378000e0 Pi*2^(1/2)/GAMMA(3/4)
3.646215960720791177099082602269212366636550840222881873870933592e1 Pi^Pi
3.739558136192022880547280543464164151116292486061500420947428024e-1 Artin
4.098748850882364744787812123379552778963580132549454698263363988e-1 HardyLittlewoodC5
4.100755656647303192888654885196002592430006070572381744864500000e-2 ZetaR(3)
4.124540336401075977833613682584552830892210999835362287235684730e-1 ThueMorse
4.146825098511116602481096221543077083657742381379169778682454145e-1 PrimesInBinary
4.210244382407083333356273792126090361362197482266604722989695515e-1 BesselK(0,1)
4.396800100000252508236009097478205164697241813798726617249562759e0 Pi*csc(5/24*Pi)/GAMMA(19/24)
4.399240125710253040409033914345447647980854079401198576534935450e-1 TreeGrowth2nd
4.400505857449335159596822037189149131273723019927652511367581718e-1 BesselJ(1,1)
4.522474200410654985065433648322479341732313432398924217364180000e-1 ZetaP(2)
4.636476090008061162142562314612144020285370542861202638109330887e-1 arctan(1/2)
4.669201609102990671853203820466201617258185577475768632745651343e0 FeigenbaumDelta
4.718616534526817848744687936113161490770126217394432436314602669e-1 Bloch
4.749493799879206503325046363279829685595493732172029822833310249e-1 Weierstrass
4.756260767359885548049772529049547142819605313538227528958111342e-1 PlouffeB
5.174790616738993863307581618988629456223774751413792582443193480e-1 polylog(4,1/2)
5.372131936080402009406232255949658266704024993403781706897619307e-1 7/8*Zeta(3)-1/12*Pi^2*log(2)+1/6*log(2)^3
5.381376357405767028067828734153656228567550149508553229391100000e-2 ZetaQ(2)
5.403023058681397174009366074429766037323104206179222276700972554e-1 cos(1)
5.432589653429767069527282953006132311388632937583569889557325691e-1 Landau
5.512212239940160755245575595481501856908419062602976216572892064e1 Psi(2,1/3)
5.566316001780235204250096895207726111398799114872853461616744626e0 2*Pi/GAMMA(5/6)
5.651591039924850272076960276098633073288996216210920094802944895e-1 BesselI(1,1)
5.671432904097838729999686622103555497538157871865125081351310792e-1 LambertW(1)
5.759599688929454396431633754924966925065139671764923636006407987e-1 Stephens
5.772156649015328606065120900824024310421593359399235988057672349e-1 gamma
5.819486593172907979281498845023675593048328730717725218234212993e-1 LandauRamanujan2nd
5.877852522924731291687059546390727685976524376431459910722724808e-1 sin(1/5*Pi)
5.926327182016361971040786049957014690842754071971610710995626082e-1 Lehmer
5.963473623231940743410784993692793760741778601525487815734849105e-1 Gompertz
6.019072301972345747375400015356173392615868899681064560177679592e-1 BesselK(1,1)
6.243299885435508709929363831008372441796426201805292869735519025e-1 GolombDickman
6.257358072052700190464746918032860288781291803142262642828972159e-1 Kac
6.351663546042712072066965912725224173420656873323724508997344605e-1 HardyLittlewoodC3
6.434105462883380261822543077575647632865878602682395059870309203e-1 Cahen
6.462454398948133042664733968457927900220129129631577293303862470e-1 MasserGramain
6.601618158468695739278121100145557784326233602847334133194484233e-1 TwinPrimes
6.617071822671762351558311332484135817464001357909536048089442295e-1 Robbin
6.627434193491815809747420971092529070562335491150224175203925350e-1 LaplaceLimit
6.931471805599453094172321214581765680755001343602552541206800095e-1 log(2)
6.975013584963659032846703508209229240731539462145153953543787529e-1 ArtinRank2
6.977746579640079820067905925517525994866582629980212323686300828e-1 BesselI(1,2)/BesselI(0,2)
7.044422009991655927366033503266372101885864314170980494142268426e-1 CareFree
7.098034428612913146417873994445755970125022057678605169570026447e-1 Rabbit
7.147827007912942720189848796210840967313455970944303193964570041e-1 TravellingSalesman
7.236484022982000094088491498091275990417837515730770291761198898e-1 Sarnak
7.281584548367672486058637587490131913773633833433795259900655974e-2 gamma(1)
7.475979202534114351787309438301781730247862640742283766042291634e-1 RenyiParking
7.642236535892206629906987312500923281167905413934095147216866737e-1 LandauRamanujan
7.651976865579665514497175261026632209092742897553252418615475491e-1 BesselJ(0,1)
7.669931397642468449426192959331578701620410597148431902649380000e-2 ZetaP(4)
7.758835100038954996204042844279006114824134659730162762210631165e-1 StronglyCareFree
7.945071927794792762403624156360456462985771009886069672658867372e-1 Kolakoski
8.090169943749474241022934171828190588601545899028814310677243114e-1 cos(1/5*Pi)
8.093940205406391307179318805940913172159539924250003042420287150e-1 AlladiGrinstead
8.125565590160063876948821016495367124344192249063615667832034758e-2 StolarskyHarborth
8.323672369053164248490954462761044873784441510122511788018155377e-1 FeigenbaumB
8.346268416740731862814297327990468089939930134903470024498273701e-1 GaussAGM
8.378669409802082408946785794357563099930066441628996920341249094e-1 Chi(1)
8.414709848078965066525023216302989996225630607983710656727517100e-1 sin(1)
8.590997968547031049035725028419742026142399555594390874354326599e-1 Thue
8.755082732970504494226765813746675051112061220425472440026000000e-3 ZetaQ(3)
8.813735870195430252326093249797923090281603282616354107532956087e-1 log(2^(1/2)+1)
8.815138397251707769283918229032278471298692572080767336701685355e-1 QuadraticClass
9.159655941772190150546035149323841107741493742816721342664981196e-1 Catalan
9.375482543158437537025740945678649778978602886148299258854334804e-1 -Zeta(1,2)
9.659258262890682867497431997288973676339048390084045504023430763e-1 cos(1/12*Pi)
9.690363192872318484530386035212529359065806101340749880701365452e-3 gamma(2)
9.864196439415648573466891734359621087334839610835655056544163385e-2 Paris

 

 

 

References


[1] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.
[2] Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 59-65, 2003.
[3] Simon Plouffe :  Plouffe Inverter : a collection of 215 million constants (1998-2006).
[4] Sloane, N. J. A.,The OEIS The On-Line Encyclopedia of Integer Sequences.
[5] Simon Plouffe :  Inverse Symbolic Calculator : a collection of 58 million constants (1995).
[6] Eric Weisstein : Mathworld
[7] François Bergeron, Simon Plouffe, Computing the Generating Function of a Series Given Its First Few Terms, Experimental Mathematics, Vol. 1 (1992), No. 4.

[8] Bruno Salvy, Paul Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software (TOMS) , Volume 20 ,  Issue 2  (June 1994), pp. 163  177.

[9] Plouffe Inverter, Version of 2005, 1.099 billion constants @ 64 digits. Not yet on the web.

[10] Xavier Gourdon and Pascal Sebah, Numbers, constants and computation.

[11] Simon Plouffe, Approximations de Séries Génératrices et quelques conjectures. Montréal, Canada: Université du Québec à Montréal, Mémoire de Maîtrise, UQAM, 1992.