OFFSET
0,3
EXAMPLE
The decimal expansion of this constant z begins:
z = 10.8363086385325049431870358764271278765976859534931...
The simple continued fraction expansion of z begins:
z = [10; 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, ..., A321097(n), ...];
such that the simple continued fraction expansion of 6*z begins:
6*z = [65; 56, 60, 64, 60, 56, 64, 60, 56, 64, 56, ..., A321097(n) + 55, ...].
EXTENDED TERMS.
The initial 1000 digits in the decimal expansion of z are
z = 10.83630863853250494318703587642712787659768595349311\
73840509756931960001734110072237735215798152265637\
46033548966251275029451235255917154869550414547346\
45601898326659069509029896766448634581870902999261\
78291099037993947368425232025742840508201019811150\
20208289541868116590985746685817208034834182741861\
61586263073936595659616093596727391439370392218179\
08547782927594504604528661115974783060857978290729\
53554586787471663938331763610007750862560295292956\
22583160832720034539915220107654291931753328805663\
44405451280922502018454665640681719991329902449206\
06333718948414803434770198192597675071144159105469\
40129387536502210902718153383173369508615022733071\
21561771111264471719424048701509094587624798702003\
98051339274318126056502629341701820569809346581703\
17900710636878987844980734936343020769115171474588\
61969660741202379814909712010859009313616125172084\
48790047790048120552938902316397984428482656427316\
35549261064509815824229787948467548551187722067240\
97844304015886508690953002055508602378218606168521...
...
The initial 1020 terms of the continued fraction of z are
z = [10;1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,
1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,
5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,
5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,
1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,
5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,
5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,
1,5,9,5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,
5,1,9,1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,1,5,9,5,1,9,
1,5,9,1,5,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9,5,1,9,5,1,9,1,5,9, ...].
...
GENERATING METHOD.
Start with CF = [10] and repeat (PARI code):
{M = contfracpnqn(CF + vector(#CF,i, 55));
z = (1/6)*M[1,1]/M[2,1]; CF = contfrac(z)}
This method can be illustrated as follows.
z0 = [10] = 10;
z1 = (1/6)*[65] = [10; 1, 5] = 65/6;
z2 = (1/6)*[65; 56, 60] = [10; 1, 5, 9, 5, 1, 9, 6] = 218525/20166;
z3 = (1/6)*[65; 56, 60, 64, 60, 56, 64, 61] = [10; 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, 5, 1, 9, 1, 5, 10] = 30617277049665/2825434201901;
z4 = (1/6)*[65; 56, 60, 64, 60, 56, 64, 60, 56, 64, 56, 60, 64, 56, 60, 64, 60, 56, 64, 56, 60, 65] = [10; 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, 5, 1, 9, 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, 5, 1, 9, 1, 5, 9, 5, 1, 9, 5, 1, 10] = 235326213809918755668077578309692661245/21716455451732827969266335806481498321;
where this constant z equals the limit of the iterations of the above process.
PROG
(PARI) /* Generate over 6000 digits */
{CF=[10]; for(i=1, 8, M = contfracpnqn( CF + vector(#CF, i, 55) ); z = (1/6)*M[1, 1]/M[2, 1]; CF = contfrac(z) )}
for(n=0, 200, print1(floor(10^(n-1)*z)%10, ", "))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Paul D. Hanna, Oct 28 2018
STATUS
approved