A320829 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, 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 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

Offset

0,2

Comments

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...

Links

Table of n, a(n) for n=0..91.

Formula

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

Example

t = 1.4467466246485661263399614145878451657650505718... (A320828);
the continued fraction of t begins
t = [1; 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, ..., floor((n+1)*t), ...].
The initial digits in the decimal expansion of t begins
t = 1.44674662464856612633996141458784516576505057180153\
38115730662100523948899875419615259508338069905046\
70892333791489618831233662716692897735289725678868\
97096617331243451184296731674644993365604464002135\
44826122090131377005103007238085314454831482149624\
85517263355852054054515598437123023419087520342074\
30641273668583671914634815049056543577672827902355\
22112737692093527826031712678252960843632048178054\
16505116090816000597993588292027144032368925698956\
49806402032615763430845499940177234220138008874302\
22704797898737162994071394166496400308279197196447\
92741983179392608019795029915894795466398263775852\
29063351986333834850687434868952422596962925772480\
60379786178748850195617564046505556222525049108813\
42022341182087931655219577768300674229035616008232\
78846346788482742187714237233512675209462092856542\
11664502047576653411225920269686880872245578761745\
66620892635287769327891323661520725226329427107814\
92834193947844880591442478232304430636356942904353\
45845611662720450849560496953804958417523863724365...

Crossrefs

Cf. A320828.

Sequence in context: A186157 A072633 A037087 * A194145 A330116 A189784

Adjacent sequences: A320826 A320827 A320828 * A320830 A320831 A320832

Keyword

nonn,cofr

Author

Paul D. Hanna, Oct 21 2018

Status

approved