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%I #9 Oct 29 2018 04:02:53
%S 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,
%T 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,
%U 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
%N Continued fraction 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.
%F Formula for terms:
%F (1) a(0) = 10,
%F (2) a(3*n) = 9 for n >= 1,
%F (3) a(3*n+2) = 6 - a(3*n+1) for n >= 0,
%F (4) a(9*n+1) = 1 for n >= 0,
%F (5) a(9*n+7) = 5 for n >= 0,
%F (6) a(9*n+4) = 6 - a(3*n+1) for n >= 0.
%F a(3*n+1) = 4*A189706(n+1) + 1 for n >= 0.
%F a(n) = 4*A321090(n) + 1 for n >= 1, with a(0) = 10.
%F a(n) = A321091(n) + 3*A321090(n), for n >= 0.
%F a(n) = A321093(n) + 2*A321090(n), for n >= 0.
%F a(n) = A321095(n) + A321090(n), for n >= 0.
%e The decimal expansion of this constant z begins:
%e z = 10.8363086385325049431870358764271278765976859534931...
%e The simple continued fraction expansion of z begins:
%e z = [10; 1, 5, 9, 5, 1, 9, 5, 1, 9, 1, 5, 9, 1, 5, 9, 5, ..., a(n), ...];
%e such that the simple continued fraction expansion of 6*z begins:
%e 6*z = [65; 56, 60, 64, 60, 56, 64, 60, 56, 64, 56, 60, ..., a(n) + 55, ...].
%e EXTENDED TERMS.
%e The initial 1020 terms of the continued fraction of z are
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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, ...].
%e ...
%e The initial 1000 digits in the decimal expansion of z are
%e z = 10.83630863853250494318703587642712787659768595349311\
%e 73840509756931960001734110072237735215798152265637\
%e 46033548966251275029451235255917154869550414547346\
%e 45601898326659069509029896766448634581870902999261\
%e 78291099037993947368425232025742840508201019811150\
%e 20208289541868116590985746685817208034834182741861\
%e 61586263073936595659616093596727391439370392218179\
%e 08547782927594504604528661115974783060857978290729\
%e 53554586787471663938331763610007750862560295292956\
%e 22583160832720034539915220107654291931753328805663\
%e 44405451280922502018454665640681719991329902449206\
%e 06333718948414803434770198192597675071144159105469\
%e 40129387536502210902718153383173369508615022733071\
%e 21561771111264471719424048701509094587624798702003\
%e 98051339274318126056502629341701820569809346581703\
%e 17900710636878987844980734936343020769115171474588\
%e 61969660741202379814909712010859009313616125172084\
%e 48790047790048120552938902316397984428482656427316\
%e 35549261064509815824229787948467548551187722067240\
%e 97844304015886508690953002055508602378218606168521...
%e ...
%e GENERATING METHOD.
%e Start with CF = [10] and repeat (PARI code):
%e {M = contfracpnqn(CF + vector(#CF,i, 55));
%e z = (1/6)*M[1,1]/M[2,1]; CF = contfrac(z)}
%e This method can be illustrated as follows.
%e z0 = [10] = 10 ;
%e z1 = (1/6)*[65] = [10; 1, 5] = 65/6 ;
%e z2 = (1/6)*[65; 56, 60] = [10; 1, 5, 9, 5, 1, 9, 6] = 218525/20166 ;
%e 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 ;
%e 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 ;
%e where this constant z equals the limit of the iterations of the above process.
%o (PARI) /* Generate over 5000 terms */
%o {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) )}
%o for(n=0,200,print1(CF[n+1],", "))
%Y Cf. A321090, A321091, A321092, A321093, A321094, A321095, A321096, A321098.
%K nonn,cofr
%O 0,1
%A _Paul D. Hanna_, Oct 28 2018
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