A321093 Continued fraction 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.

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, 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, 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, 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, 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

Offset

0,1

Links

Table of n, a(n) for n=0..200.

Formula

Formula for terms:

(1) a(0) = 6,

(2) a(3*n) = 5 for n >= 1,

(3) a(3*n+2) = 4 - a(3*n+1) for n >= 0,

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

(5) a(9*n+7) = 3 for n >= 0,

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

a(3*n+1) = 2*A189706(n+1) + 1 for n >= 0.

a(n) = 2*A321090(n) + 1 for n >= 1, with a(0) = 6.

a(n) = A321091(n) + A321090(n), for n >= 0.

a(n) = A321095(n) - A321090(n), for n >= 0.

a(n) = A321097(n) - 2*A321090(n), for n >= 0.

Example

The decimal expansion of this constant z begins:
z = 6.76134218926005257700977872218164992635251510297623...
The simple continued fraction expansion of z begins:
z = [6; 1, 3, 5, 3, 1, 5, 3, 1, 5, 1, 3, 5, 1, 3, 5, 3, ..., a(n), ...];
such that the simple continued fraction expansion of 4*z begins:
4*z = [27; 22, 24, 26, 24, 22, 26, 24, 22, 26, 22, 24, ..., a(n) + 21, ...].
EXTENDED TERMS.
The initial 1020 terms of the continued fraction of z are
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,
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,
3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
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,
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,
3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
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,
3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,
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,
3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
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,
1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,
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,
1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,
1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,
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,
1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
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,
3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,
3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,
3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,
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,
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,
3,1,5,1,3,5,1,3,5,3,1,5,1,3,5,3,1,5,3,1,5,1,3,5,1,3,5,3,1,5,
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, ...].
...
The initial 1000 digits in the decimal expansion of z are
z = 6.76134218926005257700977872218164992635251510297623\
82400543083129376225709036432953076227076440718046\
31453442451274796099781430687357859993593539615424\
85162251478631501261337528918395764663363544002143\
30566359813785491386584191187803105986019925125817\
32384559737235970178404167070340277684175375836022\
44216229551462644458639369128033581124838287016422\
22465666402121115108802903360423241018391788636516\
34399014131667135507041306777557247262858680853566\
90766431107666031620853273656348374218315881851585\
44479630341871788962324209888156077547855028352382\
55925436674100345760249501114992870759647092383954\
96007641595484979939448371742959430380908428799273\
24364811976473376482587750783735175570880836352529\
00570146062437832931408193409264288195544895081069\
61581071442858044513706835327745695285899243308879\
49562004677663100500730098181693102336025369092373\
55940511090331284958638975881509935173849203422329\
94160267792889652591286734230036989339097946760406\
04381955878548467565031395792149142079025146844422...
...
GENERATING METHOD.
Start with CF = [6] and repeat (PARI code):
{M = contfracpnqn(CF + vector(#CF,i, 21));
z = (1/4)*M[1,1]/M[2,1]; CF = contfrac(z)}
This method can be illustrated as follows.
z0 = [6] = 6 ;
z1 = (1/4)*[27] = [6; 1, 3] = 27/4 ;
z2 = (1/4)*[27; 22, 24] = [6; 1, 3, 5, 3, 1, 5, 4] = 14307/2116 ;
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 ;
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 ; ...
where this constant z equals the limit of the iterations of the above process.

Prog

(PARI) /* Generate over 5000 terms */
{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) )}
for(n=0, 200, print1(CF[n+1], ", "))

Crossrefs

Cf. A321090, A321091, A321092, A321094, A321095, A321096, A321097, A321098.

Sequence in context: A021167 A213204 A085677 * A219552 A296476 A296504

Adjacent sequences: A321090 A321091 A321092 * A321094 A321095 A321096

Keyword

nonn,cofr

Author

Paul D. Hanna, Oct 28 2018

Status

approved