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A323312 Continued fraction of 2*K where K is the constant equal to the Kolakoski sequence (A000002) when taken as a continued fraction expansion. 1
2, 1, 5, 6, 2, 2, 5, 1, 3, 2, 3, 1, 5, 2, 3, 1, 2, 1, 5, 6, 6, 2, 2, 5, 1, 3, 2, 5, 1, 2, 1, 3, 2, 3, 1, 5, 2, 2, 3, 1, 2, 1, 5, 2, 2, 3, 1, 3, 2, 5, 1, 2, 1, 3, 2, 2, 5, 1, 3, 2, 5, 1, 2, 1, 5, 6, 5, 1, 2, 1, 3, 2, 5, 1, 2, 1, 5, 6, 6, 2, 2, 3, 1, 3, 2, 2, 6, 6, 5, 1, 2, 1, 5, 6, 2, 2, 6, 5, 1, 2, 1, 5, 2, 3, 1, 3, 2, 2, 5, 1, 3, 2, 5, 1, 2, 1, 3, 2, 3, 1, 5, 2, 3, 1, 2, 1, 5, 6, 6, 2, 2, 6, 5, 1, 2, 1, 5, 6, 2, 2, 5, 1, 3, 2, 3, 1, 2, 1, 5, 6, 5, 1, 2, 1, 5, 2, 3, 1, 2, 1, 5, 6, 5, 1, 2, 1, 3, 2, 3, 1, 5, 2, 2, 6, 6, 2, 2, 3, 1, 3, 2, 5, 1, 2, 1, 5, 6, 2, 2, 6, 6, 5, 1, 2, 1, 5, 2, 3, 1, 5, 2, 2, 3, 1, 2, 1, 5, 2, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

No '4' appears to be present (checked up to 20000 terms); all terms appear to consist of only numbers [1,2,3,5,6]; the continued fraction of K/2 appears to have this same property.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..20000

EXAMPLE

Let K be the constant having a continued fraction expansion equal to the Kolakoski sequence (A000002):

K = [1; 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, ...];

K = 1.41877964149605852815364808632291708019167486692804029547087633369284...

then this sequence equals the continued fraction expansion of 2*K, where

2*K = 2.83755928299211705630729617264583416038334973385608059094175266738568...

2*K = [2; 1, 5, 6, 2, 2, 5, 1, 3, 2, 3, 1, 5, 2, 3, 1, 2, 1, 5, 6, 6, ...].

The initial 1000 terms of the continued fraction of 2*K (this sequence) starts as:

K = [2;1,5,6,2,2,5,1,3,2,3,1,5,2,3,1,2,1,5,6,6,2,2,5,1,3,

2,5,1,2,1,3,2,3,1,5,2,2,3,1,2,1,5,2,2,3,1,3,2,5,1,

2,1,3,2,2,5,1,3,2,5,1,2,1,5,6,5,1,2,1,3,2,5,1,2,1,

5,6,6,2,2,3,1,3,2,2,6,6,5,1,2,1,5,6,2,2,6,5,1,2,1,

5,2,3,1,3,2,2,5,1,3,2,5,1,2,1,3,2,3,1,5,2,3,1,2,1,

5,6,6,2,2,6,5,1,2,1,5,6,2,2,5,1,3,2,3,1,2,1,5,6,5,

1,2,1,5,2,3,1,2,1,5,6,5,1,2,1,3,2,3,1,5,2,2,6,6,2,

2,3,1,3,2,5,1,2,1,5,6,2,2,6,6,5,1,2,1,5,2,3,1,5,2,

2,3,1,2,1,5,2,2,3,1,3,2,2,6,6,2,2,5,1,3,2,5,1,2,1,

3,2,2,5,1,3,2,3,1,2,1,5,2,2,3,1,2,1,5,2,3,1,5,2,2,

6,6,2,2,3,1,3,2,2,5,1,2,1,3,2,3,1,5,2,3,1,2,1,5,2,

2,3,1,3,2,2,6,6,5,1,2,1,5,2,3,1,2,1,5,6,5,1,2,1,5,

2,3,1,5,2,2,3,1,2,1,5,2,2,3,1,3,2,2,6,6,2,2,3,1,3,

2,5,1,2,1,5,6,5,1,2,1,3,2,3,1,5,2,2,6,5,1,2,1,5,6,

2,2,6,6,5,1,2,1,5,6,2,2,5,1,3,2,3,1,2,1,5,6,5,1,2,

1,5,2,3,1,3,2,2,6,6,2,2,3,1,3,2,2,5,1,2,1,3,2,5,1,

3,2,3,1,2,1,5,2,2,3,1,3,2,2,6,6,2,2,3,1,3,2,5,1,2,

1,3,2,2,5,1,3,2,3,1,2,1,5,2,2,3,1,2,1,5,2,3,1,5,2,

2,3,1,3,2,5,1,3,2,3,1,5,2,3,1,3,2,2,5,1,3,2,5,1,2,

1,3,2,2,5,1,2,1,3,2,3,1,5,2,3,1,2,1,5,2,2,3,1,3,2,

2,6,6,2,2,3,1,3,2,5,1,2,1,3,2,2,5,1,3,2,3,1,2,1,5,

6,5,1,2,1,3,2,3,1,5,2,2,6,6,2,2,3,1,3,2,2,6,6,5,1,

2,1,5,6,2,2,6,6,5,1,2,1,3,2,5,1,3,2,3,1,5,2,2,6,6,

2,2,3,1,3,2,5,1,2,1,5,6,2,2,6,5,1,2,1,5,6,6,2,2,5,

1,3,2,5,1,2,1,3,2,3,1,5,2,2,3,1,2,1,5,2,3,1,5,2,2,

6,6,5,1,2,1,3,2,5,1,2,1,5,6,6,2,2,3,1,3,2,2,6,6,2,

2,5,1,3,2,3,1,5,2,3,1,3,2,2,5,1,2,1,3,2,2,5,1,3,2,

5,1,2,1,3,2,3,1,5,2,3,1,2,1,5,6,6,2,2,6,5,1,2,1,5,

6,2,2,5,1,3,2,3,1,2,1,5,6,5,1,2,1,5,2,3,1,2,1,5,6,

6,2,2,5,1,3,2,3,1,5,2,3,1,2,1,5,6,6,2,2,5,1,3,2,5,

1,2,1,3,2,3,1,5,2,2,6,5,1,2,1,5,6,2,2,6,6,5,1,2,1,

3,2,5,1,3,2,3,1,2,1,5,2,3,1,5,2,2,3,1,3,2,5,1,2,1,

5,6,2,2,6,5,1,2,1,5,2,3,1,2,1,5,6,5,1,2,1,3,2,3,1,

5,2,2,6,6,2,2,3,1,3,2,5,1,2,1,5,6,2,2,6,5,1,2,1,5,

6,6,2,2,5,1,3,2,5,1,2,1,3,2,3,1,5,2,2,3,1,2,1,5,2,

2,3,1,3,2,5,1,3,2,3,1,5,2,3,1,2,1,5,2,2,3,1,3,2,2,

6,6,5,1,2,1,5,2,3,1,2,1,5,6,5,1,2,1,3,2,3,1,5,2,2,

6,5,1,2,1,5,6,6,2,2,6,5,1,2,1,5,2,3,1,3,2,2,5,1,3,

2,5,1,2,1,3,2,3,1,5,2,3,1,2,1,5,6,6,2,2,5,1,3,2,3,

1,5,2,3,1,2,1,5,6,6,2,2,6,5,1,2,1,5,6,6,2,2,3,1,3,...],

and appears to consist of only numbers [1,2,3,5,6].

RELATED DECIMAL EXPANSION.

The initial 2000 digits of K are:

K = 1.41877964149605852815364808632291708019167486692804\

02954708763336928400188878238212125223580007572364\

17384329060435042278529197840919265977519727845772\

31249681924445527538269400939622941753919080178698\

44190565402841816055525264789336579398042313723735\

06894544505381199920260656532991751880179423036191\

18191781837111751310015972004338251420166916352841\

28548680352197737937586124265798291010168421108840\

71451063869739386282136133656443609202913008733448\

93977087426643496537157593270403055671400515606960\

46387972589673179715624069531153417502373099901445\

98694229073228037920174025352357836689935022884073\

14942829632338200243182971813373705320236041498261\

63725329773029816890835459547194290736121908744342\

02769094730583191437000282679742983187641917856239\

76846174791051433173202050007037234224177623710267\

83697233092721964223817503606669847565053676960085\

84525818733680602048418002414012426538439344357445\

36973349936667535562314399578485918626791470385134\

94515743336899131135946482033957425376487598552872\

37760829934688602759224332965535887302494434354329\

32811408552007902955316107872205617178536235011461\

08315328651975928447205378918900565084637355716494\

90086343112113805613214386814550534123779238004029\

55931524769449461832080202964902615444941719421242\

78580324329839092080852796747561030786671649149802\

25424769567200329860354981894175958140269364990733\

04266566455012316146828742985935394226128338546205\

00257307227211886778675098496308262187227193042845\

10938338177836227375234536174120166864707230799053\

64779272859782785249589082220162484921664015461683\

25241205459245414495193037209385728194830173951511\

03624722452966015559723383919980467050521517627260\

56712304062037479581793529463591588648735650492462\

65822702248543856728353909502671843919355228375433\

40300811312516671273697432562541372015964167798713\

87369376123590612846029906514384262681334223394506\

44915472070765873813895052158255705654501784691342\

10410008236248263787632884217448349418396431953078\

94310670012423450694219349566723654314736245884890...

PROG

(PARI) /* Informal code to generate the continued fraction of 2*K */

{A2=[1, 2, 2]; for(n=3, 2200, for(i=1, A2[n], A2=concat(A2, 2-n%2))); #A2}

PQ = contfracpnqn(A2); K = PQ[1, 1]/PQ[2, 1]; CF2=contfrac(K*2); #CF2

for(n=0, #CF2-2, print1( CF2[n+1], ", "); if(n%40==0, print("")))

CROSSREFS

Cf. A000002.

Sequence in context: A113345 A078123 A342968 * A231774 A209170 A231732

Adjacent sequences:  A323309 A323310 A323311 * A323313 A323314 A323315

KEYWORD

nonn,cofr

AUTHOR

Paul D. Hanna, Jan 17 2019

STATUS

approved

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Last modified August 1 19:59 EDT 2021. Contains 346402 sequences. (Running on oeis4.)