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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. 2
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Only integers 1..6 seem to appear in the sequence.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..65537
EXAMPLE
The continued fraction of r begins:
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
2,2,5,2,1,2,4,2,1,2,...].
The continued fraction of 2*r forms a bisection of this sequence:
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
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,
1,5,2,2,2,1,1,1,2,5,...].
The initial 2000 digits of the constant r starts
r = 2.22648946472216033467059435976265716038428808989188\
33377256285620032908005119906089940298720854948636\
10235724779998310051085960435666030670373624849860\
46488191495211979583799667285778451703768539302832\
95678669580572204615820297244741969812876059951103\
22855859549284598927656080715167410518308884880602\
81482123590537498180527127346339309551413962691546\
49180168001721243651478957770332934007750116804753\
60333252900957379539026411466558272414710112866091\
78311492707013164091221415485827348966624994611014\
97717050783144703651176548186411571117682134204981\
53459528332174688361907216540857854666249870160310\
32231051334924611466905284955424733245040746264869\
98652684460134433563003163272726681724864305400547\
76494345113374939620510204299384835402003494621561\
05768848555017525447564085021551824550695609419837\
78569833201941061645407628884426886853926164688657\
34630090951802293223239611745938539445608678110777\
98313272816661297274534214742448694455932574687116\
21751816750598941868848777268747446840802166043019\
41260887379378188453877069260666751868027107467897\
76221768885802123637720863338373619310808989283789\
61861279033048095423969203317321623313343287704471\
84686599673998326812847434349755210306672556352348\
42022537361682803283927511471370865006910207542551\
04464181731069714667005072579044559821081242186130\
88819734862428583952715416423204225684992657109265\
30972640985785340942735424728994761860283406172799\
18814617595433910289369567834685790279190616209914\
54015571336495461296493607719068521813862369672045\
32769857605512125449571060786664660248358737433169\
30459891721281050233812243615811169276355842001288\
84546465723306413930221017925863651995776811567447\
50944465954167809642385540481293254520497028604774\
83066722381090271904596595521026831381930071032672\
90902958694289594809820490646202054224860343122880\
24220853020088041254135397727063126338201524084154\
59136243499247076384671537375395407933731757156752\
42573537461650815282427757933350110087829412754349\
93918114075354470683232761482726835032441599688437...
The occurrence of 3's is rather infrequent:
a(n) = 3 when n = [79, 739, 799, 1123, 1263, 1971, 2295, 3223, 3415,
5659, 5727, 6159, 6175, 6223, 8555, 8623, 9419, 9479, 9611, 9671,
9687, 9735, 10575, 10635, 10735, 10927, 11759, 11819, 11919, 12955,
13139, 13427, 13619, 14251, 14311, 14495, 14555, 14687, 14835, 16727,
17695, 17755, 17823, 18331, 18391, 19431, 19547, 20239, 20299, 21183,
22135, 22807, 24551, 24795, 24863, 25755, 27755, 27815, 29083, 29835,
29951, 30591, 30739, 30923, 30983, 31167, 31227, 31411, 31471, 31655,
31715, 31899, 31959, 32143, 32203, 32387, 32447, 33783, 33931, 35331,
35479, 35539, 37479, 38819, 38915, 39099, 39159, 39343, 40895, 41011,
41207, 42307, 42491, 42779, 42971, 43339, 43435, 43503, 43743, 44303,
44639, 44699, 44895, 45135, 45231, 45427, 45543, 45735, 46727, 47247,
47347, 47395, 47411, 47471, 47539, 48211, 48535, 48727, 49203, 49651,
49711, 49811, 50003, 50119, 50315, ...].
PROG
(PARI) /* Generates 3, 350 terms of the continued fraction */
{A=[2]; for(i=1, 12, PQ=contfracpnqn(A); r = PQ[1, 1]/PQ[2, 1];
CF2=contfrac(2*r); A=vector(2*#CF2, n, if(n%2==1, 2, CF2[n/2])) ); }
for(n=0, 3350, print1(A[n+1], ", "))
CROSSREFS
Sequence in context: A064132 A072865 A364002 * A179686 A286479 A013604
Adjacent sequences: A322725 A322726 A322727 * A322729 A322730 A322731
KEYWORD
nonn,cofr
AUTHOR
Paul D. Hanna, Jan 31 2019
STATUS
approved

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Last modified August 18 11:16 EDT 2024. Contains 375265 sequences. (Running on oeis4.)