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
KEYWORD
nonn,cofr
AUTHOR
Paul D. Hanna, Jan 17 2019
STATUS
approved