A321096 - OEIS

%I #13 Nov 04 2018 20:35:01

%S 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,

%T 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,

%U 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

%N 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.

%e The decimal expansion of this constant z begins:

%e z = 8.80540175768866337347693999543763987641156287148453...

%e The simple continued fraction expansion of z begins:

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

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

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

%e EXTENDED TERMS.

%e The initial 1000 digits in the decimal expansion of z are

%e z = 8.80540175768866337347693999543763987641156287148453\

%e 81468516943758900177466965168383790121108844505757\

%e 17187999454407348106496636159427234274709665063530\

%e 50935777205767852706781920879673120741864445658376\

%e 11976420857048526287399704761091708055469240672330\

%e 48700326462058452646355562520962867414041721519998\

%e 23160265836475138977655743106219535120079474624210\

%e 99859736180993210725127756524124066610420161356552\

%e 06399576189405297872838825683944974598313320897473\

%e 82753910120167655227857213444857908106596044697304\

%e 03731161925824151533224356077992140610806046874738\

%e 76664918873754737673637688749508970659125480247854\

%e 23148383885317039738394794978662476138806965606055\

%e 89270492560115659978879195004407252000769277753763\

%e 08004863113948791353859078952125599689193183276991\

%e 65636567095719857927702210317282103234986565300353\

%e 38502840972838489688713570280513772584527070132759\

%e 37605094965136681249812588515170458336825974959207\

%e 93310692922707508597239859725358109680410147516880\

%e 36970398236866976270861217620395537245456726267033...

%e ...

%e The initial 1020 terms of the continued fraction of z are

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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,

%e 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, ...].

%e ...

%e GENERATING METHOD.

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

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

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

%e This method can be illustrated as follows.

%e z0 = [8] = 8 ;

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

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

%e 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 ;

%e 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 ; ...

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

%o (PARI) /* Generate over 5300 digits */

%o {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) )}

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

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

%K nonn,cons

%O 1,1

%A _Paul D. Hanna_, Oct 28 2018