login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A322728 Continued fraction expansion of a constant r such that the odd-indexed bisection equals the continued fraction of 2*r, with an even-indexed bisection of all 2's. 1

%I

%S 2,4,2,2,2,4,2,1,2,4,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,2,2,1,2,2,2,4,

%T 2,1,2,5,2,2,2,6,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,5,2,1,2,4,2,2,2,5,2,2,

%U 2,1,2,2,2,4,2,1,2,4,2,3,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,2,2,1,2,2,2,5,2,2,2,4,2,1,2,4,2,2,2,1,2,2,2,2,2,2,2,5,2,1,2,4,2,2,2,5,2,2,2,1,2

%N Continued fraction expansion of a constant r such that the odd-indexed bisection equals the continued fraction of 2*r, with an even-indexed bisection of all 2's.

%C Only integers 1..6 seem to appear in the sequence.

%e The continued fraction of r begins:

%e r = [2;4,2,2,2,4,2,1,2,4,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,2,2,1,2,

%e 2,2,4,2,1,2,5,2,2,2,6,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,5,2,1,2,

%e 4,2,2,2,5,2,2,2,1,2,2,2,4,2,1,2,4,2,3,2,4,2,1,2,4,2,1,2,4,2,

%e 1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,1,2,4,2,2,2,1,2,2,2,5,2,2,2,

%e 4,2,1,2,4,2,2,2,1,2,2,2,2,2,2,2,5,2,1,2,4,2,2,2,5,2,2,2,1,2,

%e 1,2,1,2,2,2,1,2,1,2,4,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,

%e 5,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,5,2,

%e 2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,2,2,5,2,1,2,4,2,2,2,1,2,2,2,

%e 2,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,4,2,1,2,5,2,2,2,2,2,2,2,2,2,

%e 2,2,2,2,2,2,1,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,2,2,1,2,2,2,

%e 4,2,1,2,5,2,2,2,6,2,2,2,2,2,2,2,6,2,2,2,1,2,1,2,1,2,2,2,2,2,

%e 2,2,1,2,2,2,4,2,1,2,5,2,2,2,6,2,2,2,2,2,2,2,1,2,2,2,4,2,1,2,

%e 4,2,2,2,1,2,2,2,2,2,2,2,6,2,2,2,5,2,1,2,4,2,2,2,1,2,2,2,2,2,

%e 2,2,1,2,2,2,4,2,1,2,5,2,2,2,6,2,2,2,2,2,2,2,1,2,2,2,4,2,1,2,

%e 4,2,2,2,1,2,2,2,2,2,2,2,6,2,2,2,5,2,1,2,4,2,1,2,4,2,1,2,4,2,

%e 2,2,1,2,2,2,5,2,2,2,4,2,1,2,5,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%e 6,2,2,2,5,2,1,2,4,2,2,2,1,2,2,2,2,2,2,2,1,2,1,2,1,2,2,2,5,2,

%e 2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%e 2,2,2,2,5,2,1,2,4,2,2,2,1,2,2,2,2,2,2,2,6,2,2,2,5,2,1,2,4,2,

%e 1,2,4,2,1,2,5,2,2,2,2,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,1,2,2,2,

%e 2,2,2,2,1,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,1,2,

%e 2,2,2,2,2,2,6,2,2,2,5,2,1,2,4,2,1,2,4,2,1,2,5,2,2,2,2,2,2,2,

%e 1,2,1,2,1,2,2,2,5,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,1,2,2,2,2,2,

%e 2,2,2,2,2,2,2,2,2,2,5,2,1,2,4,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,

%e 2,2,2,2,5,2,1,2,4,2,1,2,4,2,1,2,4,2,3,2,4,2,1,2,4,2,2,2,1,2,

%e 2,2,5,2,2,2,4,2,1,2,5,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,5,2,1,2,

%e 4,2,2,2,5,2,2,2,1,2,2,2,4,2,1,2,4,2,3,2,4,2,1,2,4,2,1,2,4,2,

%e 1,2,5,2,2,2,2,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,4,2,1,2,5,2,2,2,

%e 2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,5,2,

%e 2,2,1,2,1,2,1,2,2,2,5,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,2,2,

%e 5,2,1,2,4,2,2,2,1,2,2,2,2,2,2,2,1,2,1,2,1,2,2,2,5,2,2,2,1,2,

%e 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%e 1,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,5,2,2,2,1,2,1,2,1,2,2,2,2,2,

%e 2,2,5,2,1,2,4,2,1,2,...].

%e The continued fraction of 2*r forms a bisection of this sequence:

%e 2*r = [4;2,4,1,4,2,5,2,1,1,1,2,2,2,1,2,4,1,5,2,6,2,2,2,2,2,2,2,5,1,4,

%e 2,5,2,1,2,4,1,4,3,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,1,2,5,2,4,

%e 1,4,2,1,2,2,2,5,1,4,2,5,2,1,1,1,2,1,1,4,2,5,2,1,1,1,2,5,2,5,

%e 2,1,1,1,2,5,2,5,2,1,1,1,2,5,2,5,2,1,1,1,2,2,2,5,1,4,2,1,2,2,

%e 2,1,1,1,2,5,2,4,1,5,2,2,2,2,2,2,2,1,2,5,2,1,1,1,2,2,2,1,2,4,

%e 1,5,2,6,2,2,2,6,2,1,1,1,2,2,2,1,2,4,1,5,2,6,2,2,2,1,2,4,1,4,

%e 2,1,2,2,2,6,2,5,1,4,2,1,2,2,2,1,2,4,1,5,2,6,2,2,2,1,2,4,1,4,

%e 2,1,2,2,2,6,2,5,1,4,1,4,1,4,2,1,2,5,2,4,1,5,2,2,2,2,2,2,2,6,

%e 2,5,1,4,2,1,2,2,2,1,1,1,2,5,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%e 2,5,1,4,2,1,2,2,2,6,2,5,1,4,1,4,1,5,2,2,2,1,1,1,2,5,2,1,2,2,

%e 2,1,2,1,2,2,2,2,2,2,2,1,2,1,2,2,2,6,2,5,1,4,1,4,1,5,2,2,2,1,

%e 1,1,2,5,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,5,1,4,2,5,2,1,1,1,2,2,

%e 2,5,1,4,1,4,1,4,3,4,1,4,2,1,2,5,2,4,1,5,2,2,2,2,2,2,2,5,1,4,

%e 2,5,2,1,2,4,1,4,3,4,1,4,1,4,1,5,2,2,2,1,1,1,2,5,2,4,1,5,2,2,

%e 2,2,2,2,2,1,2,1,2,2,2,1,2,5,2,1,1,1,2,5,2,5,2,1,1,1,2,2,2,5,

%e 1,4,2,1,2,2,2,1,1,1,2,5,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,

%e 2,1,2,2,2,1,2,5,2,1,1,1,2,2,2,5,1,4,1,4,1,5,2,6,2,2,2,1,2,4,

%e 1,5,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,

%e 2,2,2,1,2,5,2,1,1,1,2,2,2,5,1,4,1,4,1,4,3,4,1,4,2,1,2,5,2,5,

%e 2,1,1,1,2,5,2,1,2,2,2,2,2,2,2,6,2,5,1,4,2,1,2,2,2,5,1,4,1,4,

%e 1,5,2,2,2,5,1,4,1,4,1,4,1,4,1,4,1,4,1,5,2,2,2,5,1,4,1,4,1,4,

%e 3,4,1,4,2,1,2,5,2,5,2,1,1,1,2,5,2,1,2,2,2,2,2,2,2,6,2,5,1,4,

%e 2,1,2,2,2,5,1,4,1,4,1,5,2,2,2,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,

%e 2,1,2,5,2,4,1,4,2,1,2,2,2,6,2,5,1,4,1,4,1,4,2,1,2,5,2,5,2,1,

%e 1,1,2,5,2,4,1,1,2,1,1,1,2,5,2,4,1,5,2,2,2,1,2,4,1,4,2,5,2,1,

%e 2,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,2,1,2,5,2,4,1,4,2,1,2,2,2,5,

%e 1,4,2,5,2,1,1,1,2,1,1,4,2,5,2,1,1,1,2,5,2,5,2,1,2,4,1,4,1,4,

%e 1,5,2,6,2,2,2,1,2,4,1,4,2,5,2,1,2,4,1,4,1,4,1,4,1,4,1,4,1,4,

%e 1,5,2,2,2,5,1,4,1,4,1,5,2,2,2,1,2,4,1,5,2,6,2,2,2,1,2,4,1,4,

%e 2,1,2,2,2,6,2,5,1,4,1,4,1,4,2,1,2,5,2,4,1,5,2,2,2,2,2,2,2,6,

%e 2,5,1,4,2,1,2,2,2,5,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,1,4,

%e 1,4,1,4,1,4,1,4,1,5,2,2,2,5,1,4,1,4,1,5,2,2,2,1,2,4,1,5,2,6,

%e 2,2,2,2,2,2,2,1,2,5,2,1,1,1,2,5,2,5,2,1,2,4,1,4,3,4,1,4,1,4,

%e 1,5,2,2,2,1,1,1,2,5,...].

%e The initial 2000 digits of the constant r starts

%e r = 2.22648946472216033467059435976265716038428808989188\

%e 33377256285620032908005119906089940298720854948636\

%e 10235724779998310051085960435666030670373624849860\

%e 46488191495211979583799667285778451703768539302832\

%e 95678669580572204615820297244741969812876059951103\

%e 22855859549284598927656080715167410518308884880602\

%e 81482123590537498180527127346339309551413962691546\

%e 49180168001721243651478957770332934007750116804753\

%e 60333252900957379539026411466558272414710112866091\

%e 78311492707013164091221415485827348966624994611014\

%e 97717050783144703651176548186411571117682134204981\

%e 53459528332174688361907216540857854666249870160310\

%e 32231051334924611466905284955424733245040746264869\

%e 98652684460134433563003163272726681724864305400547\

%e 76494345113374939620510204299384835402003494621561\

%e 05768848555017525447564085021551824550695609419837\

%e 78569833201941061645407628884426886853926164688657\

%e 34630090951802293223239611745938539445608678110777\

%e 98313272816661297274534214742448694455932574687116\

%e 21751816750598941868848777268747446840802166043019\

%e 41260887379378188453877069260666751868027107467897\

%e 76221768885802123637720863338373619310808989283789\

%e 61861279033048095423969203317321623313343287704471\

%e 84686599673998326812847434349755210306672556352348\

%e 42022537361682803283927511471370865006910207542551\

%e 04464181731069714667005072579044559821081242186130\

%e 88819734862428583952715416423204225684992657109265\

%e 30972640985785340942735424728994761860283406172799\

%e 18814617595433910289369567834685790279190616209914\

%e 54015571336495461296493607719068521813862369672045\

%e 32769857605512125449571060786664660248358737433169\

%e 30459891721281050233812243615811169276355842001288\

%e 84546465723306413930221017925863651995776811567447\

%e 50944465954167809642385540481293254520497028604774\

%e 83066722381090271904596595521026831381930071032672\

%e 90902958694289594809820490646202054224860343122880\

%e 24220853020088041254135397727063126338201524084154\

%e 59136243499247076384671537375395407933731757156752\

%e 42573537461650815282427757933350110087829412754349\

%e 93918114075354470683232761482726835032441599688437...

%e The occurrence of 3's is rather infrequent:

%e a(n) = 3 when n = [79, 739, 799, 1123, 1263, 1971, 2295, 3223, 3415,

%e 5659, 5727, 6159, 6175, 6223, 8555, 8623, 9419, 9479, 9611, 9671,

%e 9687, 9735, 10575, 10635, 10735, 10927, 11759, 11819, 11919, 12955,

%e 13139, 13427, 13619, 14251, 14311, 14495, 14555, 14687, 14835, 16727,

%e 17695, 17755, 17823, 18331, 18391, 19431, 19547, 20239, 20299, 21183,

%e 22135, 22807, 24551, 24795, 24863, 25755, 27755, 27815, 29083, 29835,

%e 29951, 30591, 30739, 30923, 30983, 31167, 31227, 31411, 31471, 31655,

%e 31715, 31899, 31959, 32143, 32203, 32387, 32447, 33783, 33931, 35331,

%e 35479, 35539, 37479, 38819, 38915, 39099, 39159, 39343, 40895, 41011,

%e 41207, 42307, 42491, 42779, 42971, 43339, 43435, 43503, 43743, 44303,

%e 44639, 44699, 44895, 45135, 45231, 45427, 45543, 45735, 46727, 47247,

%e 47347, 47395, 47411, 47471, 47539, 48211, 48535, 48727, 49203, 49651,

%e 49711, 49811, 50003, 50119, 50315, ...].

%o (PARI) /* Generates 3,350 terms of the continued fraction */

%o {A=[2]; for(i=1,12, PQ=contfracpnqn(A); r = PQ[1,1]/PQ[2,1];

%o CF2=contfrac(2*r); A=vector(2*#CF2,n,if(n%2==1,2,CF2[n/2])) );}

%o for(n=0,3350, print1(A[n+1],", "))

%K nonn,cofr

%O 0,1

%A _Paul D. Hanna_, Jan 31 2019

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 14 10:07 EDT 2020. Contains 335721 sequences. (Running on oeis4.)