OFFSET
0,1
FORMULA
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, 4, ..., a(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, 40, ..., a(n) + 36, ...].
EXTENDED TERMS.
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, ...].
...
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...
...
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 5000 terms */
{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=0, 200, print1(CF[n+1], ", "))
CROSSREFS
KEYWORD
nonn,cofr
AUTHOR
Paul D. Hanna, Oct 28 2018
STATUS
approved