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A321098 Decimal expansion of the constant z that satisfies: CF(6*z, n) = CF(z, n) + 55, for n >= 0, where CF(z, n) denotes the n-th partial denominator in the continued fraction expansion of z. 8
1, 0, 8, 3, 6, 3, 0, 8, 6, 3, 8, 5, 3, 2, 5, 0, 4, 9, 4, 3, 1, 8, 7, 0, 3, 5, 8, 7, 6, 4, 2, 7, 1, 2, 7, 8, 7, 6, 5, 9, 7, 6, 8, 5, 9, 5, 3, 4, 9, 3, 1, 1, 7, 3, 8, 4, 0, 5, 0, 9, 7, 5, 6, 9, 3, 1, 9, 6, 0, 0, 0, 1, 7, 3, 4, 1, 1, 0, 0, 7, 2, 2, 3, 7, 7, 3, 5, 2, 1, 5, 7, 9, 8, 1, 5, 2, 2, 6, 5, 6, 3, 7, 4, 6, 0, 3, 3, 5, 4, 8, 9, 6, 6, 2, 5, 1, 2, 7, 5, 0, 2, 9, 4, 5, 1, 2, 3, 5, 2, 5, 5, 9, 1, 7, 1, 5, 4, 8, 6, 9, 5, 5, 0, 4, 1, 4, 5, 4, 7, 3, 4, 6, 4, 5, 6, 0, 1, 8, 9, 8, 3, 2, 6, 6, 5, 9, 0, 6, 9, 5, 0, 9, 0, 2, 9, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..175.

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

Cf. A321090, A321091, A321092, A321093, A321094, A321095, A321096, A321097.

Sequence in context: A171043 A124599 A005601 * A308741 A304580 A304583

Adjacent sequences:  A321095 A321096 A321097 * A321099 A321100 A321101

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Oct 28 2018

STATUS

approved

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Last modified December 13 06:26 EST 2019. Contains 329968 sequences. (Running on oeis4.)