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%I #14 Nov 01 2018 06:47:13
%S 6,7,6,1,3,4,2,1,8,9,2,6,0,0,5,2,5,7,7,0,0,9,7,7,8,7,2,2,1,8,1,6,4,9,
%T 9,2,6,3,5,2,5,1,5,1,0,2,9,7,6,2,3,8,2,4,0,0,5,4,3,3,3,6,5,6,0,3,3,9,
%U 3,1,4,6,7,6,9,0,0,4,3,1,7,5,0,5,5,8,4,3,9,3,5,4,2,7,0,8,5,9,2,3,5,9,2,2,7,8,6,9,0,3,4,1,1,6,1,5,4,2,0,9,6,7,6,6,6,3,7,5,7,9,8,5,3,1,7,7,3,6,9,3,4,3,5,1,9,0,0,7,8,1,2,0,3,9
%N Decimal expansion of the constant z that satisfies: CF(4*z, n) = CF(z, n) + 21, 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 = 6.76134218926005257700977872218164992635251510297623...
%e The simple continued fraction expansion of z begins:
%e z = [6; 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, ..., A321093(n), ...];
%e such that the simple continued fraction expansion of 4*z begins:
%e 4*z = [27; 22, 24, 26, 24, 22, 26, 24, 22, 26, 22, 24, ..., A321093(n) + 21, ...].
%e EXTENDED TERMS.
%e The initial 1000 digits in the decimal expansion of z are
%e z = 6.76134218926005257700977872218164992635251510297623\
%e 82400543083129376225709036432953076227076440718046\
%e 31453442451274796099781430687357859993593539615424\
%e 85162251478631501261337528918395764663363544002143\
%e 30566359813785491386584191187803105986019925125817\
%e 32384559737235970178404167070340277684175375836022\
%e 44216229551462644458639369128033581124838287016422\
%e 22465666402121115108802903360423241018391788636516\
%e 34399014131667135507041306777557247262858680853566\
%e 90766431107666031620853273656348374218315881851585\
%e 44479630341871788962324209888156077547855028352382\
%e 55925436674100345760249501114992870759647092383954\
%e 96007641595484979939448371742959430380908428799273\
%e 24364811976473376482587750783735175570880836352529\
%e 00570146062437832931408193409264288195544895081069\
%e 61581071442858044513706835327745695285899243308879\
%e 49562004677663100500730098181693102336025369092373\
%e 55940511090331284958638975881509935173849203422329\
%e 94160267792889652591286734230036989339097946760406\
%e 04381955878548467565031395792149142079025146844422...
%e ...
%e The initial 1020 terms of the continued fraction of z are
%e z = [6;1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
%e 1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
%e 3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
%e 3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
%e 1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
%e 3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
%e 3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
%e 1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,
%e 3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
%e 1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5, ...].
%e ...
%e GENERATING METHOD.
%e Start with CF = [6] and repeat (PARI code):
%e {M = contfracpnqn(CF + vector(#CF,i, 21));
%e z = (1/4)*M[1,1]/M[2,1]; CF = contfrac(z)}
%e This method can be illustrated as follows.
%e z0 = [6] = 6;
%e z1 = (1/4)*[27] = [6; 1, 3] = 27/4;
%e z2 = (1/4)*[27; 22, 24] = [6; 1, 3, 5, 3, 1, 5, 4] = 14307/2116;
%e z3 = (1/4)*[27; 22, 24, 26, 24, 22, 26, 25] = [6; 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, 1, 5, 1, 3, 6] = 32184125853/4760020267;
%e z4 = (1/4)*[27; 22, 24, 26, 24, 22, 26, 24, 22, 26, 22, 24, 26, 22, 24, 26, 24, 22, 26, 22, 24, 27] = [6; 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, 1, 5, 1, 3, 5, 3, 1, 5, 3, 1, 6] = 668028962666848193442388706037/98801235607919425997683216483; ...
%e where this constant z equals the limit of the iterations of the above process.
%o (PARI) /* Generate over 4600 digits */
%o {CF=[6]; for(i=1,8, M = contfracpnqn( CF + vector(#CF,i, 21) ); z = (1/4)*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, A321095, A321096, A321097, A321098.
%K nonn,cons
%O 1,1
%A _Paul D. Hanna_, Oct 28 2018