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Continued fraction of the positive constant t in (1,2) such that the partial denominators form the Beatty sequence {floor((n+1)*t), n >= 0}.
1

%I #16 Oct 10 2019 04:10:48

%S 1,2,4,5,7,8,10,11,13,14,15,17,18,20,21,23,24,26,27,28,30,31,33,34,36,

%T 37,39,40,41,43,44,46,47,49,50,52,53,54,56,57,59,60,62,63,65,66,67,69,

%U 70,72,73,75,76,78,79,81,82,83,85,86,88,89,91,92,94,95,96,98,99,101,102,104,105,107,108,109,111,112,114,115,117,118,120,121,122,124,125,127,128,130,131,133

%N Continued fraction of the positive constant t in (1,2) such that the partial denominators form the Beatty sequence {floor((n+1)*t), n >= 0}.

%C There exists a unique value r = r(m) in (m,m+1) such that the partial denominators of the continued fraction of r equals {floor((n+1)*r), n >= 0}, where this constant t equals r(1); r(0) = 0.70871657065865538045295674204934626302195740088521664571...

%F a(n) = floor((n+1)*t), where t = [1; 2, 4, 5, 7, 8, ... ,floor((n+1)*t), ...], for n >= 0.

%e t = 1.4467466246485661263399614145878451657650505718... (A320828);

%e the continued fraction of t begins

%e t = [1; 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, ..., floor((n+1)*t), ...].

%e The initial digits in the decimal expansion of t begins

%e t = 1.44674662464856612633996141458784516576505057180153\

%e 38115730662100523948899875419615259508338069905046\

%e 70892333791489618831233662716692897735289725678868\

%e 97096617331243451184296731674644993365604464002135\

%e 44826122090131377005103007238085314454831482149624\

%e 85517263355852054054515598437123023419087520342074\

%e 30641273668583671914634815049056543577672827902355\

%e 22112737692093527826031712678252960843632048178054\

%e 16505116090816000597993588292027144032368925698956\

%e 49806402032615763430845499940177234220138008874302\

%e 22704797898737162994071394166496400308279197196447\

%e 92741983179392608019795029915894795466398263775852\

%e 29063351986333834850687434868952422596962925772480\

%e 60379786178748850195617564046505556222525049108813\

%e 42022341182087931655219577768300674229035616008232\

%e 78846346788482742187714237233512675209462092856542\

%e 11664502047576653411225920269686880872245578761745\

%e 66620892635287769327891323661520725226329427107814\

%e 92834193947844880591442478232304430636356942904353\

%e 45845611662720450849560496953804958417523863724365...

%Y Cf. A320828.

%K nonn,cofr

%O 0,2

%A _Paul D. Hanna_, Oct 21 2018