A321095 - OEIS

%I #12 Oct 29 2018 04:00:48

%S 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,

%T 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,

%U 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

%N Continued fraction 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.

%F Formula for terms:

%F (1) a(0) = 8,

%F (2) a(3*n) = 7 for n >= 1,

%F (3) a(3*n+2) = 5 - a(3*n+1) for n >= 0,

%F (4) a(9*n+1) = 1 for n >= 0,

%F (5) a(9*n+7) = 4 for n >= 0,

%F (6) a(9*n+4) = 5 - a(3*n+1) for n >= 0.

%F a(3*n+1) = 3*A189706(n+1) + 1 for n >= 0.

%F a(n) = 3*A321090(n) + 1 for n >= 1, with a(0) = 8.

%F a(n) = A321091(n) + 2*A321090(n), for n >= 0.

%F a(n) = A321093(n) + A321090(n), for n >= 0.

%F a(n) = A321097(n) - A321090(n), for n >= 0.

%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, 4, ..., a(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, 40, ..., a(n) + 36, ...].

%e EXTENDED TERMS.

%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 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 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 5000 terms */

%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=0,200,print1(CF[n+1],", "))

%Y Cf. A321090, A321091, A321092, A321093, A321094, A321096, A321097, A321098.

%K nonn,cofr

%O 0,1

%A _Paul D. Hanna_, Oct 28 2018