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Table of convergents constants

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This page is based on empirical evidence.

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If you use the convergents of the simple continued fraction of a positive real constant 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 .

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

Decimal expansion


0.555753104278045912445404118914340954558701121527873520909466749141805525[1]
3/2[2]
2.34840747027923017753942106197568446599459134194436379240686093933819431
3.27650338501442446313869723500191021836425538416806540917422208480175505
4.22347020820989381899229465579606520590540168426834718889825404922428100
5.18565646992802409017974449702978508707523321914128178742372087927236700
6.15810303145254827410063597755065162511167287820224098634493149065143559
7.13735628625292704311007524679169877304322715574918267136020828074914214
8.12126618497817841315615757896912822918546207374829639453021008588079632
9.10846444629835855991014157961308368936311583840647537012962893409038950
10.0980577066244279660274026688371803570441462414026209663468193828590617
11.0894425212553401078876965434845548031348191003781045829218889648259711
12.0821993897558878542198280177025526317026280405662984566729881856315719
13.0760285296874645132629224136174909740168070390661300623065877509924028
14.0707106421306583794307985590019607518185910897586656901770854255534492
15.0660818368963185795758515870216283423710967675570285878485746555729410
16.0620173529872245808707445573926099244176869445335986119694645662767158
17.0584206140878372258151041771708503107019010736610995430331403081780732
18.0552157567107736203683948969249555639263462699212290812321530303026009
19.0523423877911115732097644781511097132248947281357809368283750631322836
20.0497518564660224543274614039363517238124817233619002142629711440569487
21.0474045440909825260028174458150879195859547035079107915457441450023133
22.0452678714518992721450037899058489734476536614104800501601558782252837
23.0433148070044688113813661667828989390215175430801967752526064653220158
24.0415227390704340349445242968831853205024363492002894981264231200960497
25.0398726104835531505454800119187743870010480391023212748288096665713029
26.0383482489519995759396825385239952339632533596060289949098856476522737
27.0369358423734693183441925373183671665720756853453752356678870073522203
28.0356235247014360291341001488753207326643401862148362051108062748636572
29.0344010455745866474204942206947531129812752761653152146317756652818656
30.0332595050811042418532747973329799470071726245407823526215258495081172
31.0321911388527757111622591442900622936149620560651599893433295038856995
32.0311891429623599978929347293525123501331510248600142086752535560797423
33.0302475301065362835343522296099366839415836523882970866444110641167673
34.0293610109002658556654484410723038745374048555565585493442508693343757
35.0285248952067097223198351872514511326533377515569370270226176388988650
36.0277350097606288916277724739781161105813921494296070098554776932025361
37.0269876289651056822609144095977817853723011681638548196504164332390172
38.0262794165266744637155697353909617372129300610278021133239820782856376
39.0256073759572507636965033170757647816824074805674443381026310239717574
40.0249688084476915260996790257012337366038359258054184341836304371630757
41.0243612768360387748560272092573627265904058368454691615018168095940850
42.0237825746904769269490776383145522691082581957635278361925790992943925
43.0232306996614274856554962343968451184394725443237088202658570830949899
44.0227038304468644021082316993531158741502891356873352360321218985462016
45.0222003067995898679269732517860875121318662297622698424543079083633103
46.0217186121290207012920572164987690595482794100502596494486142804546010
47.0212573583044634217851372765589486255486837426359011649361616200947105
48.0208152723493080992155772873888601237085126185045255822258948589065609
49.0203911847512214370852112126080912169085890115746720120719388884542624
50.0199840191693372283284223648026071485989281172741672014671750209923297
51.0195927833431935961128410190667810389646441105501495457647763617903339
52.0192165610467264846727952191413248629001860113327586169807435780044138
53.0188545049467010030371106812489064513716149387900157099230325907091442
54.0185058302519633905613316789928150712664625089867550993597550319032703
55.0181698090509289822945925561213673297969398552911031312215028514016890
56.0178457652538979500217427030043169137434640573203488862506542974062819
57.0175330700644644713389562472396524966345130726179705154995937790676919
58.0172311379180834749302762609310234471323921478605500706933066161279209
59.0169394228312622851524064536129272388599209711261731951376550771828887
60.0166574151148944999937393782499861437550594465363278073750638942404134
61.0163846384091010451838019305685685026390053035487426888784292269388758
62.0161206470043462795337138560989980380842851030966535182523422843536872
63.0158650234163658085747332323737806149454324381956602033203091485212667
64.0156173761879534390703539567572611559989327718459899341097879456344791
65.0153773378926666897183344770173211437629562604281687262136027337233822
66.0151445633196530806147717923102208305152003068353702012892760476065051
67.0149187278202748180991130155244737432479173630206519026868590545271941
68.0146995258003527442229053334812513702429592488423029206966337882944678
69.0144866693429415491575106836010008301920634946690326478418495360084892
70.0142798869489538291983317437338963015332271892547871421040190535511643
71.0140789223837639406078897520375739540234815447014145045836192858086094
72.0138835336197786707289784137324307272398273925714541550178370280324108
73.0136934918655725746514755250863726286010647704569040678434471403798632
74.0135085806736289266802832650346609755343779339181036035231393699217408
75.0133285951191891540469524020312468944131977208162821381408353551779317
76.0131533410438437787131485095352602919999478360136113363781711061779171
77.0129826343578494844092271147092362172357949240559711594806786360030450
78.0128163003960480748175902186222641585116833493333494356095658896616408
79.0126541733225323421733485650628379625011304899781880029744658044224723
80.0124960955799111121269738637889217436022267168601969869780015858734444
81.0123419173792331121758392012538763800427384168408191102171862996337661
82.0121914962271940593415929774424133239468846960029179503334293354439164
83.0120446964874119499341124542092520390357653331327374115684092632463724
84.0119013889730091283550356079183166160637452196762106637426400397593614
85.0117614505678646915825930276524547374643012972636073211990740398374752
86.0116247638742671247476388760829830756973667215199145048290535917347995
87.0114912168847947913328696552970936804891529990008771871083447599505842
88.0113607026765493318071122213830776657576432122905651667796101017122746
89.0112331191259437822358472195463418349782782747284205505747284268540866
90.0111083686424899224423031240518536485406496536639501758085933324906162
91.0109863579200899007682860129730159145331268043813004142685990178451639
92.0108669977045361698439668170097885118553846556725829901809638858006542
93.0107502025759716979075200073193106499789867282803181797087724353927445
94.0106358907452263164678839837347951325471865228968205287383862491574476
95.0105239838629831508204537512064265082040532946087762147584694952969109
96.0104144068408646641532393224365069927441978848755228825134768041837314
97.0103070876835582111443789477402482269771953477158977238816425204544377
98.0102019573312136294970474488006649950020753495440125744008421101479272
99.0100989495113696831238805448930028090438552837401407240721717463482767
100.009998000599760111943429983564997375509180836539380597659977240111413
1.50000000000000000000000000000000000
10.09805770662442796602740266883718036
100.00999800059976011194342998356499738
1000.00099999800000599997600011199943400
10000.00009999999800000005999999760000011
100000.00000999999999800000000059999999976
1.00000000000099999999999800000000000600000*10^6
1.000000000000009999999999999800000000000006*10^7
1.0000000000000000999999999999999800000000000*10^8
1.00000000000000000099999999999999999800000000*10^9
1.000000000000000000009999999999999999999800000*10^10
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 and the first few partial denominators 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

Integer part and

partial denominators


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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
3 2 6 1 9 1 12 1 16 1 20 1 24 1 28 1 32 1 36 1
1 4 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 3 1
2 11 2 1 4 2 58 3 2 4 1 5 110 6 2 7 1 8 163 9
1 389 2 1 3 3 9 4 1 5 2 7 9 8 1 9 2 11 2 12
2 1 1 1 4 6 2 2 7 1 9 7 2 2 13 1 15 7 2 2
4 5 1 4 1 1 2 8 1 4 1 1 3 6 1 3 1 1 3 5
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 and the partial denominators of the simple continued fractions of the convergents constants seem to follow the pattern





.




As (is this for the unknown behaviors of both and ?)


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 , and for . — 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

The first few convergents are

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

In general, we would expect that

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

Now we're at the case

Substituting , above, the next iteration is equal to

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

Therefore in general, the next coefficient should be

Now the analysis splits into two cases, whether , is even or odd. You can get , this way. Since , 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 . However, the heuristic estimates I use should give you the value of all coefficients "for large " – 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 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.