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%I #18 Jan 01 2020 22:06:24
%S 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,
%T 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,
%U 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
%N Continued fraction of 2*K where K is the constant equal to the Kolakoski sequence (A000002) when taken as a continued fraction expansion.
%C 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.
%H Paul D. Hanna, <a href="/A323312/b323312.txt">Table of n, a(n) for n = 0..20000</a>
%e Let K be the constant having a continued fraction expansion equal to the Kolakoski sequence (A000002):
%e K = [1; 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, ...];
%e K = 1.41877964149605852815364808632291708019167486692804029547087633369284...
%e then this sequence equals the continued fraction expansion of 2*K, where
%e 2*K = 2.83755928299211705630729617264583416038334973385608059094175266738568...
%e 2*K = [2; 1, 5, 6, 2, 2, 5, 1, 3, 2, 3, 1, 5, 2, 3, 1, 2, 1, 5, 6, 6, ...].
%e The initial 1000 terms of the continued fraction of 2*K (this sequence) starts as:
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,...],
%e and appears to consist of only numbers [1,2,3,5,6].
%e RELATED DECIMAL EXPANSION.
%e The initial 2000 digits of K are:
%e K = 1.41877964149605852815364808632291708019167486692804\
%e 02954708763336928400188878238212125223580007572364\
%e 17384329060435042278529197840919265977519727845772\
%e 31249681924445527538269400939622941753919080178698\
%e 44190565402841816055525264789336579398042313723735\
%e 06894544505381199920260656532991751880179423036191\
%e 18191781837111751310015972004338251420166916352841\
%e 28548680352197737937586124265798291010168421108840\
%e 71451063869739386282136133656443609202913008733448\
%e 93977087426643496537157593270403055671400515606960\
%e 46387972589673179715624069531153417502373099901445\
%e 98694229073228037920174025352357836689935022884073\
%e 14942829632338200243182971813373705320236041498261\
%e 63725329773029816890835459547194290736121908744342\
%e 02769094730583191437000282679742983187641917856239\
%e 76846174791051433173202050007037234224177623710267\
%e 83697233092721964223817503606669847565053676960085\
%e 84525818733680602048418002414012426538439344357445\
%e 36973349936667535562314399578485918626791470385134\
%e 94515743336899131135946482033957425376487598552872\
%e 37760829934688602759224332965535887302494434354329\
%e 32811408552007902955316107872205617178536235011461\
%e 08315328651975928447205378918900565084637355716494\
%e 90086343112113805613214386814550534123779238004029\
%e 55931524769449461832080202964902615444941719421242\
%e 78580324329839092080852796747561030786671649149802\
%e 25424769567200329860354981894175958140269364990733\
%e 04266566455012316146828742985935394226128338546205\
%e 00257307227211886778675098496308262187227193042845\
%e 10938338177836227375234536174120166864707230799053\
%e 64779272859782785249589082220162484921664015461683\
%e 25241205459245414495193037209385728194830173951511\
%e 03624722452966015559723383919980467050521517627260\
%e 56712304062037479581793529463591588648735650492462\
%e 65822702248543856728353909502671843919355228375433\
%e 40300811312516671273697432562541372015964167798713\
%e 87369376123590612846029906514384262681334223394506\
%e 44915472070765873813895052158255705654501784691342\
%e 10410008236248263787632884217448349418396431953078\
%e 94310670012423450694219349566723654314736245884890...
%o (PARI) /* Informal code to generate the continued fraction of 2*K */
%o {A2=[1, 2, 2]; for(n=3, 2200, for(i=1, A2[n], A2=concat(A2, 2-n%2))); #A2}
%o PQ = contfracpnqn(A2); K = PQ[1,1]/PQ[2,1]; CF2=contfrac(K*2); #CF2
%o for(n=0,#CF2-2, print1( CF2[n+1],", "); if(n%40==0,print("")))
%Y Cf. A000002.
%K nonn,cofr
%O 0,1
%A _Paul D. Hanna_, Jan 17 2019
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