A320828 Decimal expansion of the positive constant t in (1,2) such that the partial denominators of the continued fraction of t equals the Beatty sequence of t, {floor((n+1)*t), n >= 0}.

1, 4, 4, 6, 7, 4, 6, 6, 2, 4, 6, 4, 8, 5, 6, 6, 1, 2, 6, 3, 3, 9, 9, 6, 1, 4, 1, 4, 5, 8, 7, 8, 4, 5, 1, 6, 5, 7, 6, 5, 0, 5, 0, 5, 7, 1, 8, 0, 1, 5, 3, 3, 8, 1, 1, 5, 7, 3, 0, 6, 6, 2, 1, 0, 0, 5, 2, 3, 9, 4, 8, 8, 9, 9, 8, 7, 5, 4, 1, 9, 6, 1, 5, 2, 5, 9, 5, 0, 8, 3, 3, 8, 0, 6, 9, 9, 0, 5, 0, 4, 6, 7, 0, 8, 9, 2, 3, 3, 3, 7, 9, 1, 4, 8, 9, 6, 1, 8, 8, 3, 1, 2, 3, 3, 6, 6, 2, 7, 1, 6, 6, 9, 2, 8, 9, 7, 7, 3, 5, 2, 8, 9, 7, 2, 5, 6, 7, 8, 8, 6

Offset

1,2

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=1..150.

Example

t = 1.4467466246485661263399614145878451657650505718...;
the continued fraction of t (A320829) 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...

Prog

(PARI) /* Informal code to generate 1000 digits of the constant */
\p1010
t=sqrt(2)
for(i=1, 20, CF = vector(1001, n, floor(n*t)); t = contfracpnqn(CF)[1, 2]/contfracpnqn(CF)[2, 2]*1.); t

Crossrefs

Cf. A320829.

Sequence in context: A262260 A203632 A278766 * A065677 A006672 A107287

Adjacent sequences: A320825 A320826 A320827 * A320829 A320830 A320831

Keyword

nonn,cons

Author

Paul D. Hanna, Oct 21 2018

Status

approved