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Table of convergents constants
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 .
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.
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.
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} |
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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
- ↑ 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.
- ↑ Filmus, Yuval (2011-06-20). “Is the Iterated Continued fraction from Convergents for Pi/2 exactly 3/2? (answer#1)”. [math.stackexchange]. Retrieved 2011-06-21.