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Table of convergents constants

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If you use the convergents of the simple continued fraction of a positive real constant \scriptstyle x \, as the terms of a generalized continued fraction, then likewise use the new convergents in another generalized continued fraction, then repeat that process ad infinitum, you get the convergents constant of \scriptstyle x \,.

Contents

Table of decimal expansions of the convergents constants

Here is a table of the decimal expansions of the convergents constants (to a few decimal places) of numbers in unit intervals.


Decimal expansions of the convergents constants
Interval

\scriptstyle n \,<\, x \,<\, n+1 \,

Decimal expansion


\scriptstyle 0\,< \,x\,< \,1\, 0.555753104278045912445404118914340954558701121527873520909466749141805525[1]
\scriptstyle 1\,< \,x\,< \,2\, 3/2[2]
\scriptstyle 2\,< \,x\,< \,3\, 2.34840747027923017753942106197568446599459134194436379240686093933819431
\scriptstyle 3\,< \,x\,< \,4\, 3.27650338501442446313869723500191021836425538416806540917422208480175505
\scriptstyle 4\,< \,x\,< \,5\, 4.22347020820989381899229465579606520590540168426834718889825404922428100
\scriptstyle 5\,< \,x\,< \,6\, 5.18565646992802409017974449702978508707523321914128178742372087927236700
\scriptstyle 6\,< \,x\,< \,7\, 6.15810303145254827410063597755065162511167287820224098634493149065143559
\scriptstyle 7\,< \,x\,< \,8\, 7.13735628625292704311007524679169877304322715574918267136020828074914214
\scriptstyle 8\,< \,x\,< \,9\, 8.12126618497817841315615757896912822918546207374829639453021008588079632
\scriptstyle 9\,< \,x\,< \,10\, 9.10846444629835855991014157961308368936311583840647537012962893409038950
\scriptstyle 10\,< \,x\,< \,11\, 10.0980577066244279660274026688371803570441462414026209663468193828590617
\scriptstyle 11\,< \,x\,< \,12\, 11.0894425212553401078876965434845548031348191003781045829218889648259711
\scriptstyle 12\,< \,x\,< \,13\, 12.0821993897558878542198280177025526317026280405662984566729881856315719
\scriptstyle 13\,< \,x\,< \,14\, 13.0760285296874645132629224136174909740168070390661300623065877509924028
\scriptstyle 14\,< \,x\,< \,15\, 14.0707106421306583794307985590019607518185910897586656901770854255534492
\scriptstyle 15\,< \,x\,< \,16\, 15.0660818368963185795758515870216283423710967675570285878485746555729410
\scriptstyle 16\,< \,x\,< \,17\, 16.0620173529872245808707445573926099244176869445335986119694645662767158
\scriptstyle 17\,< \,x\,< \,18\, 17.0584206140878372258151041771708503107019010736610995430331403081780732
\scriptstyle 18\,< \,x\,< \,19\, 18.0552157567107736203683948969249555639263462699212290812321530303026009
\scriptstyle 19\,< \,x\,< \,20\, 19.0523423877911115732097644781511097132248947281357809368283750631322836
\scriptstyle 20\,< \,x\,< \,21\, 20.0497518564660224543274614039363517238124817233619002142629711440569487
\scriptstyle 21\,< \,x\,< \,22\, 21.0474045440909825260028174458150879195859547035079107915457441450023133
\scriptstyle 22\,< \,x\,< \,23\, 22.0452678714518992721450037899058489734476536614104800501601558782252837
\scriptstyle 23\,< \,x\,< \,24\, 23.0433148070044688113813661667828989390215175430801967752526064653220158
\scriptstyle 24\,< \,x\,< \,25\, 24.0415227390704340349445242968831853205024363492002894981264231200960497
\scriptstyle 25\,< \,x\,< \,26\, 25.0398726104835531505454800119187743870010480391023212748288096665713029
\scriptstyle 26\,< \,x\,< \,27\, 26.0383482489519995759396825385239952339632533596060289949098856476522737
\scriptstyle 27\,< \,x\,< \,28\, 27.0369358423734693183441925373183671665720756853453752356678870073522203
\scriptstyle 28\,< \,x\,< \,29\, 28.0356235247014360291341001488753207326643401862148362051108062748636572
\scriptstyle 29\,< \,x\,< \,30\, 29.0344010455745866474204942206947531129812752761653152146317756652818656
\scriptstyle 30\,< \,x\,< \,31\, 30.0332595050811042418532747973329799470071726245407823526215258495081172
\scriptstyle 31\,< \,x\,< \,32\, 31.0321911388527757111622591442900622936149620560651599893433295038856995
\scriptstyle 32\,< \,x\,< \,33\, 32.0311891429623599978929347293525123501331510248600142086752535560797423
\scriptstyle 33\,< \,x\,< \,34\, 33.0302475301065362835343522296099366839415836523882970866444110641167673
\scriptstyle 34\,< \,x\,< \,35\, 34.0293610109002658556654484410723038745374048555565585493442508693343757
\scriptstyle 35\,< \,x\,< \,36\, 35.0285248952067097223198351872514511326533377515569370270226176388988650
\scriptstyle 36\,< \,x\,< \,37\, 36.0277350097606288916277724739781161105813921494296070098554776932025361
\scriptstyle 37\,< \,x\,< \,38\, 37.0269876289651056822609144095977817853723011681638548196504164332390172
\scriptstyle 38\,< \,x\,< \,39\, 38.0262794165266744637155697353909617372129300610278021133239820782856376
\scriptstyle 39\,< \,x\,< \,40\, 39.0256073759572507636965033170757647816824074805674443381026310239717574
\scriptstyle 40\,< \,x\,< \,41\, 40.0249688084476915260996790257012337366038359258054184341836304371630757
\scriptstyle 41\,< \,x\,< \,42\, 41.0243612768360387748560272092573627265904058368454691615018168095940850
\scriptstyle 42\,< \,x\,< \,43\, 42.0237825746904769269490776383145522691082581957635278361925790992943925
\scriptstyle 43\,< \,x\,< \,44\, 43.0232306996614274856554962343968451184394725443237088202658570830949899
\scriptstyle 44\,< \,x\,< \,45\, 44.0227038304468644021082316993531158741502891356873352360321218985462016
\scriptstyle 45\,< \,x\,< \,46\, 45.0222003067995898679269732517860875121318662297622698424543079083633103
\scriptstyle 46\,< \,x\,< \,47\, 46.0217186121290207012920572164987690595482794100502596494486142804546010
\scriptstyle 47\,< \,x\,< \,48\, 47.0212573583044634217851372765589486255486837426359011649361616200947105
\scriptstyle 48\,< \,x\,< \,49\, 48.0208152723493080992155772873888601237085126185045255822258948589065609
\scriptstyle 49\,< \,x\,< \,50\, 49.0203911847512214370852112126080912169085890115746720120719388884542624
\scriptstyle 50\,< \,x\,< \,51\, 50.0199840191693372283284223648026071485989281172741672014671750209923297
\scriptstyle 51\,< \,x\,< \,52\, 51.0195927833431935961128410190667810389646441105501495457647763617903339
\scriptstyle 52\,< \,x\,< \,53\, 52.0192165610467264846727952191413248629001860113327586169807435780044138
\scriptstyle 53\,< \,x\,< \,54\, 53.0188545049467010030371106812489064513716149387900157099230325907091442
\scriptstyle 54\,< \,x\,< \,55\, 54.0185058302519633905613316789928150712664625089867550993597550319032703
\scriptstyle 55\,< \,x\,< \,56\, 55.0181698090509289822945925561213673297969398552911031312215028514016890
\scriptstyle 56\,< \,x\,< \,57\, 56.0178457652538979500217427030043169137434640573203488862506542974062819
\scriptstyle 57\,< \,x\,< \,58\, 57.0175330700644644713389562472396524966345130726179705154995937790676919
\scriptstyle 58\,< \,x\,< \,59\, 58.0172311379180834749302762609310234471323921478605500706933066161279209
\scriptstyle 59\,< \,x\,< \,60\, 59.0169394228312622851524064536129272388599209711261731951376550771828887
\scriptstyle 60\,< \,x\,< \,61\, 60.0166574151148944999937393782499861437550594465363278073750638942404134
\scriptstyle 61\,< \,x\,< \,62\, 61.0163846384091010451838019305685685026390053035487426888784292269388758
\scriptstyle 62\,< \,x\,< \,63\, 62.0161206470043462795337138560989980380842851030966535182523422843536872
\scriptstyle 63\,< \,x\,< \,64\, 63.0158650234163658085747332323737806149454324381956602033203091485212667
\scriptstyle 64\,< \,x\,< \,65\, 64.0156173761879534390703539567572611559989327718459899341097879456344791
\scriptstyle 65\,< \,x\,< \,66\, 65.0153773378926666897183344770173211437629562604281687262136027337233822
\scriptstyle 66\,< \,x\,< \,67\, 66.0151445633196530806147717923102208305152003068353702012892760476065051
\scriptstyle 67\,< \,x\,< \,68\, 67.0149187278202748180991130155244737432479173630206519026868590545271941
\scriptstyle 68\,< \,x\,< \,69\, 68.0146995258003527442229053334812513702429592488423029206966337882944678
\scriptstyle 69\,< \,x\,< \,70\, 69.0144866693429415491575106836010008301920634946690326478418495360084892
\scriptstyle 70\,< \,x\,< \,71\, 70.0142798869489538291983317437338963015332271892547871421040190535511643
\scriptstyle 71\,< \,x\,< \,72\, 71.0140789223837639406078897520375739540234815447014145045836192858086094
\scriptstyle 72\,< \,x\,< \,73\, 72.0138835336197786707289784137324307272398273925714541550178370280324108
\scriptstyle 73\,< \,x\,< \,74\, 73.0136934918655725746514755250863726286010647704569040678434471403798632
\scriptstyle 74\,< \,x\,< \,75\, 74.0135085806736289266802832650346609755343779339181036035231393699217408
\scriptstyle 75\,< \,x\,< \,76\, 75.0133285951191891540469524020312468944131977208162821381408353551779317
\scriptstyle 76\,< \,x\,< \,77\, 76.0131533410438437787131485095352602919999478360136113363781711061779171
\scriptstyle 77\,< \,x\,< \,78\, 77.0129826343578494844092271147092362172357949240559711594806786360030450
\scriptstyle 78\,< \,x\,< \,79\, 78.0128163003960480748175902186222641585116833493333494356095658896616408
\scriptstyle 79\,< \,x\,< \,80\, 79.0126541733225323421733485650628379625011304899781880029744658044224723
\scriptstyle 80\,< \,x\,< \,81\, 80.0124960955799111121269738637889217436022267168601969869780015858734444
\scriptstyle 81\,< \,x\,< \,82\, 81.0123419173792331121758392012538763800427384168408191102171862996337661
\scriptstyle 82\,< \,x\,< \,83\, 82.0121914962271940593415929774424133239468846960029179503334293354439164
\scriptstyle 83\,< \,x\,< \,84\, 83.0120446964874119499341124542092520390357653331327374115684092632463724
\scriptstyle 84\,< \,x\,< \,85\, 84.0119013889730091283550356079183166160637452196762106637426400397593614
\scriptstyle 85\,< \,x\,< \,86\, 85.0117614505678646915825930276524547374643012972636073211990740398374752
\scriptstyle 86\,< \,x\,< \,87\, 86.0116247638742671247476388760829830756973667215199145048290535917347995
\scriptstyle 87\,< \,x\,< \,88\, 87.0114912168847947913328696552970936804891529990008771871083447599505842
\scriptstyle 88\,< \,x\,< \,89\, 88.0113607026765493318071122213830776657576432122905651667796101017122746
\scriptstyle 89\,< \,x\,< \,90\, 89.0112331191259437822358472195463418349782782747284205505747284268540866
\scriptstyle 90\,< \,x\,< \,91\, 90.0111083686424899224423031240518536485406496536639501758085933324906162
\scriptstyle 91\,< \,x\,< \,92\, 91.0109863579200899007682860129730159145331268043813004142685990178451639
\scriptstyle 92\,< \,x\,< \,93\, 92.0108669977045361698439668170097885118553846556725829901809638858006542
\scriptstyle 93\,< \,x\,< \,94\, 93.0107502025759716979075200073193106499789867282803181797087724353927445
\scriptstyle 94\,< \,x\,< \,95\, 94.0106358907452263164678839837347951325471865228968205287383862491574476
\scriptstyle 95\,< \,x\,< \,96\, 95.0105239838629831508204537512064265082040532946087762147584694952969109
\scriptstyle 96\,< \,x\,< \,97\, 96.0104144068408646641532393224365069927441978848755228825134768041837314
\scriptstyle 97\,< \,x\,< \,98\, 97.0103070876835582111443789477402482269771953477158977238816425204544377
\scriptstyle 98\,< \,x\,< \,99\, 98.0102019573312136294970474488006649950020753495440125744008421101479272
\scriptstyle 99\,< \,x\,< \,100\, 99.0100989495113696831238805448930028090438552837401407240721717463482767
\scriptstyle 100\,< \,x\,< \,101\, 100.009998000599760111943429983564997375509180836539380597659977240111413
\scriptstyle 1\,< \,x\,< \,2\, 1.50000000000000000000000000000000000
\scriptstyle 10\,< \,x\,< \,11\, 10.09805770662442796602740266883718036
\scriptstyle 100\,< \,x\,< \,101\, 100.00999800059976011194342998356499738
\scriptstyle 1000\,< \,x\,< \,1001\, 1000.00099999800000599997600011199943400
\scriptstyle 10000\,< \,x\,< \,10001\, 10000.00009999999800000005999999760000011
\scriptstyle 100000\,< \,x\,< \,100001\, 100000.00000999999999800000000059999999976
\scriptstyle 1000000\,< \,x\,< \,1000001\, 1.00000000000099999999999800000000000600000*10^6
\scriptstyle 10000000\,< \,x\,< \,10000001\, 1.000000000000009999999999999800000000000006*10^7
\scriptstyle 100000000\,< \,x\,< \,100000001\, 1.0000000000000000999999999999999800000000000*10^8
\scriptstyle 1000000000\,< \,x\,< \,1000000001\, 1.00000000000000000099999999999999999800000000*10^9
\scriptstyle 10000000000\,< \,x\,< \,10000000001\, 1.000000000000000000009999999999999999999800000*10^10
\scriptstyle 100000000000\,< \,x\,< \,100000000001\, 1.0000000000000000000000999999999999999999999800*10^11

To greater precision that term for the last row, 10^11 to 10^11+1, reveals the following pattern.

1

00000000000.000000000009999999999999999999998

00000000000000000000059999999999999999999976

00000000000000000001119999999999999999999434

0000000000000000002999999999999999999998355599999999976

0000009242400000000275999994696799999999620800003089932

0000002755199981725123999997143973344235799022223768553919


Marvin Ray Burns 17:55, 8 June 2011 (UTC)

Table of integer part and partial denominators of the convergents constants

Here is a table of the integer part \scriptstyle a_0(n) \, and the first few partial denominators \scriptstyle a_i(n),\, i \,\ge\, 1, \, of the simple continued fractions of the convergents constants of numbers in unit intervals.


Integer part and partial denominators of the simple continued fractions of the convergents constants
Interval

\scriptstyle n \,<\, x \,<\, n+1 \,

Integer part \scriptstyle (a_0(n)) \, and

partial denominators \scriptstyle (a_i(n),\, i \,\ge\, 1) \,


\scriptstyle \{a_0(n),\, a_1(n),\, a_2(n),\, \ldots\} \,

\scriptstyle 0\,< \,x\,< \,1\, ?
\scriptstyle 1\,< \,x\,< \,2\, [1; 2]
\scriptstyle 2\,< \,x\,< \,3\, {2,2,1,6,1,2,2,1,1,1,1,1,1,1,2}
\scriptstyle 3\,< \,x\,< \,4\, {3,3,1,1,1,1,1,1,4,3,1,1,14,1,16}
\scriptstyle 4\,< \,x\,< \,5\, {4,4,2,9,2,4,3,4,1,2,7,1,1,63,13}
\scriptstyle 5\,< \,x\,< \,6\, {5,5,2,1,1,2,3,6,1,14,1,49,1,114,2}
\scriptstyle 6\,< \,x\,< \,7\, {6,6,3,12,1,58,9,2,2,1,3,3,1,2,18}
\scriptstyle 7\,< \,x\,< \,8\, {7,7,3,1,1,3,4,2,8,1,5,53,1,4,1}
\scriptstyle 8\,< \,x\,< \,9\, {8,8,4,16,1,2,1,7,1,2,7,1,2,7,6}
\scriptstyle 9\,< \,x\,< \,10\, {9,9,4,1,1,4,5,1,4,15,1,2,40,16,2}
\scriptstyle 10\,< \,x\,< \,11\, {10,10,5,20,1,1,2,9,1,17,1,1,1,5,10}
\scriptstyle 11\,< \,x\,< \,12\, {11,11,5,1,1,5,7,7,1,1,5,1,1,4,2}
\scriptstyle 12\,< \,x\,< \,13\, {12,12,6,24,2,110,9,2,3,1,32,1,2,1,19}
\scriptstyle 13\,< \,x\,< \,14\, {13,13,6,1,1,6,8,2,6,3,10,1,3,1,1}
\scriptstyle 14\,< \,x\,< \,15\, {14,14,7,28,2,2,1,13,1,1,6,1,3,2,2}
\scriptstyle 15\,< \,x\,< \,16\, {15,15,7,1,1,7,9,1,3,1,6,1,1,1,9}
\scriptstyle 16\,< \,x\,< \,17\, {16,16,8,32,2,1,2,15,1,5,2,1,3,18,5}
\scriptstyle 17\,< \,x\,< \,18\, {17,17,8,1,1,8,11,7,1,11,5,1,1,5,1}
\scriptstyle 18\,< \,x\,< \,19\, {18,18,9,36,3,163,2,2,3,3,1,7,4,19,1}
\scriptstyle 19\,< \,x\,< \,20\, {19,19,9,1,1,9,12,2,5,2,20,4,3,3,8}
\scriptstyle 20\,< \,x\,< \,21\, {20,20,10,40,3,2,1,19,2,9,3,6,1,1,1}
\scriptstyle 21\,< \,x\,< \,22\, {21,21,10,1,1,10,13,1,3,1,3,2,1,1,1}
\scriptstyle 22\,< \,x\,< \,23\, {22,22,11,44,3,1,2,21,1,3,1,7,2,1,2}
\scriptstyle 23\,< \,x\,< \,24\, {23,23,11,1,1,11,15,8,6,1,10,1,1,3,3}
\scriptstyle 24\,< \,x\,< \,25\, {24,24,12,48,4,217,16,1,1,1,4,1,9,6,10}
\scriptstyle 25\,< \,x\,< \,26\, {25,25,12,1,1,12,16,2,5,11,5,8,4,8,2}
\scriptstyle 26\,< \,x\,< \,27\, {26,26,13,52,4,2,1,25,2,3,1,4,2,10,1}
\scriptstyle 27\,< \,x\,< \,28\, {27,27,13,1,1,13,17,1,3,1,2,2,273,2,25}
\scriptstyle 28\,< \,x\,< \,29\, {28,28,14,56,4,1,2,27,1,3,3,3,1,1,9}
\scriptstyle 29\,< \,x\,< \,30\, {29,29,14,1,1,14,19,8,3,2,2,17,1,1,7}
\scriptstyle 30\,< \,x\,< \,31\, {30,30,15,60,5,270,1,5,1,1,1,1,2,1,4}
\scriptstyle 31\,< \,x\,< \,32\, {31,31,15,1,1,15,20,2,4,1,6,25,1,3,9}
\scriptstyle 32\,< \,x\,< \,33\, {32,32,16,64,5,2,1,31,2,2,1,2,3,1,1}
\scriptstyle 33\,< \,x\,< \,34\, {33,33,16,1,1,16,21,1,3,1,2,10,1,3,4}
\scriptstyle 34\,< \,x\,< \,35\, {34,34,17,68,5,1,2,33,1,2,1,656,7,6,1}
\scriptstyle 35\,< \,x\,< \,36\, {35,35,17,1,1,17,23,8,2,1,1,49,1,30,1}
\scriptstyle 36\,< \,x\,< \,37\, {36,36,18,72,6,324,1,2,2,2,1,1,1,2,3}
\scriptstyle 37\,< \,x\,< \,38\, {37,37,18,1,1,18,24,2,4,1,2,2,2,2,5}
\scriptstyle 38\,< \,x\,< \,39\, {38,38,19,76,6,2,1,37,2,2,4,1,2,2,5}
\scriptstyle 39\,< \,x\,< \,40\, {39,39,19,1,1,19,25,1,3,1,1,1,6,8,1}
\scriptstyle 40\,< \,x\,< \,41\, {40,40,20,80,6,1,2,39,1,2,1,4,4,2,2}
\scriptstyle 41\,< \,x\,< \,42\, {41,41,20,1,1,20,27,8,2,10,1,1,10,2,2}
\scriptstyle 42\,< \,x\,< \,43\, {42,42,21,84,7,378,1,1,1,1,5,2,1,1,1}
\scriptstyle 43\,< \,x\,< \,44\, {43,43,21,1,1,21,28,2,4,1,1,1,1,28,1}
\scriptstyle 44\,< \,x\,< \,45\, {44,44,22,88,7,2,1,43,2,1,1,16,1,1,1}
\scriptstyle 45\,< \,x\,< \,46\, {45,45,22,1,1,22,29,1,3,1,1,1,2,2,2}
\scriptstyle 46\,< \,x\,< \,47\, {46,46,23,92,7,1,2,45,1,2,1,2,8,2,1}
\scriptstyle 47\,< \,x\,< \,48\, {47,47,23,1,1,23,31,8,1,1,6,6,5,2,2}
\scriptstyle 48\,< \,x\,< \,49\, {48,48,24,96,8,432,1,1,7,1,1,2,1,2,16}
\scriptstyle 49\,< \,x\,< \,50\, {49,49,24,1,1,24,32,2,4,1,1,10,2,1,2}
\scriptstyle 50\,< \,x\,< \,51\, {50,50,25,100,8,2,1,49,2,1,1,3,2,1,1}
\scriptstyle 51\,< \,x\,< \,52\, {51,51,25,1,1,25,33,1,3,1,1,1,1,1,1}
\scriptstyle 52\,< \,x\,< \,53\, {52,52,26,104,8,1,2,51,1,2,1,1,2,2,2}
\scriptstyle 53\,< \,x\,< \,54\, {53,53,26,1,1,26,35,8,1,1,2,2,3,19,14}
\scriptstyle 54\,< \,x\,< \,55\, {54,54,27,108,9,486,2,8,2,3,7,1,1,2,6}
\scriptstyle 55\,< \,x\,< \,56\, {55,55,27,1,1,27,36,2,4,2,6,5,10,1,2}
\scriptstyle 56\,< \,x\,< \,57\, {56,56,28,112,9,2,1,55,2,1,1,1,1,7,1}
\scriptstyle 57\,< \,x\,< \,58\, {57,57,28,1,1,28,37,1,3,1,1,1,1,10,3}
\scriptstyle 58\,< \,x\,< \,59\, {58,58,29,116,9,1,2,57,1,2,1,1,16,3,1}
\scriptstyle 59\,< \,x\,< \,60\, {59,59,29,1,1,29,39,8,1,1,1,1,1,19,28}
\scriptstyle 60\,< \,x\,< \,61\, {60,60,30,120,10,540,2,2,1,4,1,3,5,1,1}
\scriptstyle 61\,< \,x\,< \,62\, {61,61,30,1,1,30,40,2,4,2,2,2,3,3,1}
\scriptstyle 62\,< \,x\,< \,63\, {62,62,31,124,10,2,1,61,2,1,1,1,3,3,1}
\scriptstyle 63\,< \,x\,< \,64\, {63,63,31,1,1,31,41,1,3,1,1,2,6,4,1}
\scriptstyle 64\,< \,x\,< \,65\, {64,64,32,128,10,1,2,63,1,2,2,5,1,1,1}



Integer part and partial denominators of the simple continued fractions of the convergents constants
\scriptstyle a_i(n),\, i \,\ge\, 0 \, \scriptstyle n \,=\, 0 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 1 \,

\scriptstyle n \,=\, 1 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 2 \,

\scriptstyle n \,=\, 2 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 3 \,

\scriptstyle n \,=\, 3 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 4 \,

\scriptstyle n \,=\, 4 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 5 \,

\scriptstyle n \,=\, 5 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 6 \,

\scriptstyle n \,=\, 6 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 7 \,

\scriptstyle n \,=\, 7 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 8 \,

\scriptstyle n \,=\, 8 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 9 \,

\scriptstyle n \,=\, 9 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 10 \,

\scriptstyle n \,=\, 10 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 11 \,

\scriptstyle n \,=\, 11 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 12 \,

\scriptstyle n \,=\, 12 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 13 \,

\scriptstyle n \,=\, 13 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 14 \,

\scriptstyle n \,=\, 14 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 15 \,

\scriptstyle n \,=\, 15 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 16 \,

\scriptstyle n \,=\, 16 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 17 \,

\scriptstyle n \,=\, 17 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 18 \,

\scriptstyle n \,=\, 18 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 19 \,

\scriptstyle n \,=\, 19 \,

\scriptstyle \,<\, x \,<\, \,

\scriptstyle 20 \,

\scriptstyle a_0(n) \, 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
\scriptstyle a_1(n) \, 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
\scriptstyle a_2(n) \, 1 2 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
\scriptstyle a_3(n) \, 3 2 6 1 9 1 12 1 16 1 20 1 24 1 28 1 32 1 36 1
\scriptstyle a_4(n) \, 1 4 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 3 1
\scriptstyle a_5(n) \, 2 11 2 1 4 2 58 3 2 4 1 5 110 6 2 7 1 8 163 9
\scriptstyle a_6(n) \, 1 389 2 1 3 3 9 4 1 5 2 7 9 8 1 9 2 11 2 12
\scriptstyle a_7(n) \, 2 1 1 1 4 6 2 2 7 1 9 7 2 2 13 1 15 7 2 2
\scriptstyle a_8(n) \, 4 5 1 4 1 1 2 8 1 4 1 1 3 6 1 3 1 1 3 5
\scriptstyle a_9(n) \, 14 6 1 3 2 14 1 1 2 15 17 1 1 3 1 1 5 11 3 2


Conjectured pattern

The integer part \scriptstyle a_0(n) \, and the partial denominators \scriptstyle a_i(n),\, i \,\ge\, 1, \, of the simple continued fractions of the convergents constants seem to follow the pattern

a_0(n) = n, \quad n \ge 0; \,


a_1(n) = \left\{     
\begin{array}{lr}       
n+1, \quad 0 \le n \le 1, \\  
n, \quad n \ge 2; \\     
\end{array} \right. \,


a_2(n) = \Bigg\lfloor{\frac{n}{2}}\Bigg\rfloor, \quad n \ge 2; \,


a_3(n) = 1 + \Bigg(\frac{1+(-1)^n}{2}\Bigg) ~ (2n-1), \quad n \ge 6; \,


 a_4(n) = 1 + \Bigg(\frac{1+(-1)^n}{2}\Bigg) ~ \Bigg(\Bigg\lfloor{\frac{n}{6}}\Bigg\rfloor - 1\Bigg), \quad n \ge 6; \,.


a_5(n) = \left\{     
\begin{array}{lr}
\left\{     
\begin{array}{lr}       
9n+4, \quad 1 < n \le 6, \\  
9n+2, \quad 6 < n \le 12, \\  
9n+1, \quad 12 < n \le 24, \\  
9n+0, \quad 24 < n \le \ ?, \\     
\end{array} \right\}, \quad n ~\bmod~ 6 = 0, \\
\\ 
\frac{n-1}{2}, \quad n ~\bmod~ 6 = 1, \\
~~~2, \quad n ~\bmod~ 6 = 2, \\
\frac{n-1}{2}, \quad n ~\bmod~ 6 = 3, \\
~~~1, \quad n ~\bmod~ 6 = 4, \\
\frac{n-1}{2}, \quad n ~\bmod~ 6 = 5; \\    
\end{array} \right. \,


a_6(n) = \left\{
\begin{array}{lr}
\left\{     
\begin{array}{lr} 
~~~~~~\lfloor \frac{2n}{3} \rfloor, \quad n ~\bmod~ 6 = 5, \\   
~~~~~~~~~~2, \quad n ~\bmod~ 6 = 4, \\ 
\,{\lfloor \frac{2n}{3} \rfloor} - 1, \quad n ~\bmod~ 6 = 3, \\ 
~~~~~~~~~~1, \quad n ~\bmod~ 6 = 2, \\ 
~~~~~~\lfloor \frac{2n}{3} \rfloor, \quad n ~\bmod~ 6 = 1, \\
\\

\left\{  
\begin{array}{lr}      
a_6(0)  = 1 \\
a_6(6)  = 9 \\ 
a_6(12) = 9 \\
a_6(18) = 2 \\
a_6(24) = 16 \\    
\end{array} \right\},\quad n ~\bmod~ 6 = 0, \\

\end{array} \right\},\quad 0 \le n \le 24 \\
\\ 
\left\{     
\begin{array}{lr}  
~~~~~~\lfloor \frac{2n}{3} \rfloor, \quad n ~\bmod~ 6 = 5, \\   
~~~~~~~~~~2, \quad n ~\bmod~ 6 = 4, \\ 
\,{\lfloor \frac{2n}{3} \rfloor} - 1, \quad n ~\bmod~ 6 = 3, \\ 
~~~~~~~~~~1, \quad n ~\bmod~ 6 = 2, \\ 
~~~~~~\lfloor \frac{2n}{3} \rfloor, \quad n ~\bmod~ 6 = 1, \\
\\
~~~~\lfloor \frac{n-1}{24} \rfloor, \quad n ~\bmod~ 6 = 0, \\
\end{array} \right\}, \quad 24 < n \le ~?, \\
\end{array} \right. \,


As \scriptstyle n \,\to\, \infty, ~? \, (is this for the unknown behaviors of both \scriptstyle a_5 \, and \scriptstyle a_6 \,?)


Marvin Ray Burns 01:00, 30 May 2011 (UTC)

[LaTeX conversion by Daniel Forgues 01:32, 9 June 2011 (UTC)]

Partial proof of pattern

Yuval Filmus at http://math.stackexchange.com/questions/39981/extract-a-pattern-of-iterated-continued-fractions-from-convergents wrote the following that as far as I can tell confirms that pattern, in general, for \scriptstyle a_0,\, a_1 \,, and \scriptstyle a_3 \, for \scriptstyle n > 2 \,. — Marvin Ray Burns 18:00, 8 June 2011 (UTC)


START[ Here is some analysis for the actual definition.

Suppose that the original continued fraction is

[a; b, c, d, \ldots]. \,

The first few convergents are

\{a, a + 1/b, a + 1/(b + 1/c), \ldots\}. \,

Therefore, the continued fraction with convergents as coefficients is equal to

a + 1/(a + 1/b + 1/(a + 1/(b + 1/c) + \cdots)). \,

In general, we would expect that

1/b + 1/(a + \cdots) < 1, \,

this will happen eventually. In that case, we can recover the second coefficient of the continued fraction as \scriptstyle a \,.

Now we're at the case

[a; a, c, d, \ldots]. \,

Substituting \scriptstyle b \,=\, a \,, above, the next iteration is equal to

a + 1/(a + 1/a + 1/(a + 1/(a + 1/c) + \cdots)). \,

Let's express that as an integral continued fraction. After peeling off the first two coefficients, we are left with

 \frac{1}{1/a + 1/(a + 1/(a + 1/c))} \approx \frac{1}{1/a + 1/a} = a/2. \,

Therefore in general, the next coefficient should be \scriptstyle \lfloor a/2 \rfloor. \,

Now the analysis splits into two cases, whether \scriptstyle a \,, is even or odd. You can get \scriptstyle a_3,\, a_4 \,, this way. Since \scriptstyle a_4 \,, involves division by 6; we know have 6 cases. And so on.

In order to prove that the process almost always converges to the constant, one needs to be more careful and show that the estimates above are mostly true. Probably one can get some conditions on the original continued fraction, and deduce from them that convergence happens "for most values", with some precise meaning.

This analysis will also help explain why you get different behavior for small \scriptstyle n \,. However, the heuristic estimates I use should give you the value of all coefficients "for large \scriptstyle n \," – how large depends on the actual coefficient. END]


[I did some presentation edits on the above — Daniel Forgues 00:07, 9 June 2011 (UTC)]


Notes

  1. Marvin Ray Burns' experiments have indicated that for \scriptstyle 0\,< \,x\,< \,1\, if an iterate>1 and is not an integer, no other iterate will be an integer and the convergents constant(cc) will be the same as other cc's with the same integral value of that iterate. If an iterate ever becomes an integer the cc will be that integer. Else the cc is 0.555753104278045912445404118914340954558701121527873520909466749141805525.... See [[For 0<x<1]] in discussion.
  2. Filmus, Yuval (2011-06-20). "Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2? (answer#1)". [math.stackexchange]. Retrieved 2011-06-21. 
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