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A321096 Decimal expansion of the constant z that satisfies: CF(5*z, n) = CF(z, n) + 36, for n >= 0, where CF(z, n) denotes the n-th partial denominator in the continued fraction expansion of z. 8
8, 8, 0, 5, 4, 0, 1, 7, 5, 7, 6, 8, 8, 6, 6, 3, 3, 7, 3, 4, 7, 6, 9, 3, 9, 9, 9, 5, 4, 3, 7, 6, 3, 9, 8, 7, 6, 4, 1, 1, 5, 6, 2, 8, 7, 1, 4, 8, 4, 5, 3, 8, 1, 4, 6, 8, 5, 1, 6, 9, 4, 3, 7, 5, 8, 9, 0, 0, 1, 7, 7, 4, 6, 6, 9, 6, 5, 1, 6, 8, 3, 8, 3, 7, 9, 0, 1, 2, 1, 1, 0, 8, 8, 4, 4, 5, 0, 5, 7, 5, 7, 1, 7, 1, 8, 7, 9, 9, 9, 4, 5, 4, 4, 0, 7, 3, 4, 8, 1, 0, 6, 4, 9, 6, 6, 3, 6, 1, 5, 9, 4, 2, 7, 2, 3, 4, 2, 7, 4, 7, 0, 9, 6, 6, 5, 0, 6, 3, 5, 3, 0, 5, 0, 9, 3, 5, 7, 7, 7, 2, 0, 5, 7, 6, 7, 8, 5, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..168.

EXAMPLE

The decimal expansion of this constant z begins:

z = 8.80540175768866337347693999543763987641156287148453...

The simple continued fraction expansion of z begins:

z = [8; 1, 4, 7, 4, 1, 7, 4, 1, 7, 1, 4, 7, 1, 4, 7, ..., A321095(n), ...];

such that the simple continued fraction expansion of 5*z begins:

5*z = [44; 37, 40, 43, 40, 37, 43, 40, 37, 43, 37, ..., A321095(n) + 36, ...].

EXTENDED TERMS.

The initial 1000 digits in the decimal expansion of z are

z = 8.80540175768866337347693999543763987641156287148453\

81468516943758900177466965168383790121108844505757\

17187999454407348106496636159427234274709665063530\

50935777205767852706781920879673120741864445658376\

11976420857048526287399704761091708055469240672330\

48700326462058452646355562520962867414041721519998\

23160265836475138977655743106219535120079474624210\

99859736180993210725127756524124066610420161356552\

06399576189405297872838825683944974598313320897473\

82753910120167655227857213444857908106596044697304\

03731161925824151533224356077992140610806046874738\

76664918873754737673637688749508970659125480247854\

23148383885317039738394794978662476138806965606055\

89270492560115659978879195004407252000769277753763\

08004863113948791353859078952125599689193183276991\

65636567095719857927702210317282103234986565300353\

38502840972838489688713570280513772584527070132759\

37605094965136681249812588515170458336825974959207\

93310692922707508597239859725358109680410147516880\

36970398236866976270861217620395537245456726267033...

...

The initial 1020 terms of the continued fraction of z are

z = [8;1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,

1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,

4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,

4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,

1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,

4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,

4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,

1,4,7,4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,

4,1,7,1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,1,4,7,4,1,7,

1,4,7,1,4,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7,4,1,7,4,1,7,1,4,7, ...].

...

GENERATING METHOD.

Start with CF = [8] and repeat (PARI code):

{M = contfracpnqn(CF + vector(#CF,i, 36));

z = (1/5)*M[1,1]/M[2,1]; CF = contfrac(z)}

This method can be illustrated as follows.

z0 = [8] = 8 ;

z1 = (1/5)*[44] = [8; 1, 4] = 44/5 ;

z2 = (1/5)*[44; 37, 40] = [8; 1, 4, 7, 4, 1, 7, 5] = 65204/7405 ;

z3 = (1/5)*[44; 37, 40, 43, 40, 37, 43, 41] = [8; 1, 4, 7, 4, 1, 7, 4, 1, 7, 1, 4, 7, 1, 4, 7, 4, 1, 7, 1, 4, 8] = 1467584898352/166668703909 ;

z4 = (1/5)*[44; 37, 40, 43, 40, 37, 43, 40, 37, 43, 37, 40, 43, 37, 40, 43, 40, 37, 43, 37, 40, 44] = [8; 1, 4, 7, 4, 1, 7, 4, 1, 7, 1, 4, 7, 1, 4, 7, 4, 1, 7, 1, 4, 7, 1, 4, 7, 4, 1, 7, 1, 4, 7, 4, 1, 7, 4, 1, 7, 1, 4, 7, 4, 1, 7, 4, 1, 7, 1, 4, 7, 1, 4, 7, 4, 1, 7, 1, 4, 7, 4, 1, 7, 4, 1, 8] = 38573876143771692243421005778572392/4380705980858842541559248009960149 ; ...

where this constant z equals the limit of the iterations of the above process.

PROG

(PARI) /* Generate over 5300 digits */

{CF=[8]; for(i=1, 8, M = contfracpnqn( CF + vector(#CF, i, 36) ); z = (1/5)*M[1, 1]/M[2, 1]; CF = contfrac(z) )}

for(n=1, 200, print1(floor(10^(n-1)*z)%10, ", "))

CROSSREFS

Cf. A321090, A321091, A321092, A321093, A321094, A321095, A321097, A321098.

Sequence in context: A231097 A309820 A309826 * A243508 A144802 A275477

Adjacent sequences:  A321093 A321094 A321095 * A321097 A321098 A321099

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Oct 28 2018

STATUS

approved

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Last modified June 17 12:26 EDT 2021. Contains 345080 sequences. (Running on oeis4.)