A320413 - OEIS

A320413

Decimal expansion of the constant t having the continued fraction expansion {d(n), n>=0} such that the continued fraction expansion of 4*t yields partial denominators {7*d(n), n>=0}.

1, 7, 8, 5, 4, 7, 4, 5, 6, 0, 9, 7, 4, 0, 6, 2, 5, 7, 5, 6, 5, 6, 0, 4, 4, 8, 0, 7, 8, 9, 4, 4, 7, 0, 2, 3, 7, 1, 7, 0, 4, 0, 1, 3, 4, 6, 1, 9, 4, 7, 1, 5, 7, 8, 3, 8, 6, 4, 2, 7, 9, 8, 2, 0, 7, 5, 8, 0, 3, 9, 9, 5, 7, 8, 5, 4, 8, 1, 6, 5, 1, 1, 5, 1, 2, 2, 0, 6, 9, 7, 3, 2, 8, 3, 7, 0, 5, 0, 4, 9, 0, 6, 1, 2, 8, 6, 4, 0, 4, 6, 3, 5, 7, 0, 5, 0, 7, 7, 1, 0, 2, 7, 7, 4, 5, 4, 7, 2, 4, 9, 6, 1, 0, 7, 5, 1, 3, 6, 7, 0

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Is this constant transcendental?

Compare to the continued fraction expansions of sqrt(3) and 3*sqrt(3), which are related by a factor of 5: sqrt(3) = [1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] and 3*sqrt(3) = [5; 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, ...].

Further, let CF(x) denote the simple continued fraction expansion of x, then we have the related identities which hold for n >= 1:

(C1) CF( (4*n+1) * sqrt((n+1)/n) ) = (4*n+3) * CF( sqrt((n+1)/n) ),

(C2) CF( (2*n+1) * sqrt((n+2)/n) ) = (2*n+3) * CF( sqrt((n+2)/n) ).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..5000

FORMULA

Given t = [a(0); a(1), a(2), a(3), a(4), a(5), a(6), ...], some related simple continued fractions are:

(1) 4*t = [7*a(0); 7*a(1), 7*a(2), 7*a(3), 7*a(4), 7*a(5), ...],

(2) 4*t/7 = [a(0); 49*a(1), a(2), 49*a(3), a(4), 49*a(5), a(6), ...],

(3) 28*t = [49*a(0); a(1), 49*a(2), a(3), 49*a(4), a(5), 49*a(6), ...].

EXAMPLE

The decimal expansion of this constant t begins:

t = 1.785474560974062575656044807894470237170401346194715783864279820...

The simple continued fraction expansion of t begins:

t = [1; 1, 3, 1, 1, 1, 20, 1, 1, 1, 3, 1, 1, 28, 35, 28, 1, 1, 3, 1, 1, 1, 20, 1, 1, 1, 3, 1, 1, 784, 61, 3, 1, 48, 3, 1, 1, 28, ..., a(n), ...]

such that the simple continued fraction expansion of 4*t begins:

4*t = [7; 7, 21, 7, 7, 7, 140, 7, 7, 7, 21, 7, 7, 196, 245, 196, 7, 7, 21, 7, 7, 7, 140, 7, 7, 7, 21, 7, 7, 5488, 427, 21, 7, 679, ..., 7*a(n), ...].

...

The initial 1000 digits in the decimal expansion of t are

t = 1.78547456097406257565604480789447023717040134619471\

57838642798207580399578548165115122069732837050490\

61286404635705077102774547249610751367094378190709\

85401921611227678484369568036324073754957652390489\

39943379031303125762694739475382462408109374813120\

00422151267123870355135847303104193193077189589829\

99475276214685582004074592162502265622639066962207\

90264315701034911225864026241008924068247829601272\

51264292894203497339763733333382142446328347420764\

99374490081465556514449152015450022286517069550755\

66197498206618148152838524399347921835127003551875\

27576125361547533763998734682666496571601524379160\

71274633004761476783978729230894485114372110439235\

32434589015973787010030486968307426525808424138019\

30747406263726641419798279952039943508589013753829\

08707915177794914067867811104635718728142844757658\

92545040853582478172771767319724386460332134939668\

53646394220175623167681377901091602489813663749106\

20051189089949357240864658051397802563633845923264\

45649305634826816712996792415779406659039071052772...

...

The initial 528 terms in the continued fraction expansion of t are

t = [1;1,3,1,1,1,20,1,1,1,3,1,1,28,35,28,1,1,3,1,1,1,20,1,1,

1,3,1,1,784,61,3,1,48,3,1,1,28,5,3,1,1,28,1,1,3,34,1,3,

1,1,1,6,1,1,1,3,1,4,1,1,6,1,1,1,3,1,1371,3,1,106,84,1,

1,3,83,1,3,5,28,1,1,3,48,1,3,8,1,1,20,1,1,1,3,1,1,784,

1,1,3,1,1,1,20,1,1,59,28,5,3,1,1,28,1,1,3,10,3,1,1,28,

1,1,3,5,28,7,28,1,1,3,10,3,1,1,28,1,1,3,5,28,2399,3,1,4,

1,1,6,1,1,185,2352,1,1,3,1,1,1,20,1,1,144,1,3,1,1,1,20,1,

1,8,3,1,48,3,1,1,28,5,3,1,83,3,1,1,84,14,28,1,1,3,34,1,

3,1,1,1,6,1,1,1,3,1,4,1,1,6,1,1,1,3,1,1371,3,1,1,28,

5,3,1,1,28,1,1,3,34,1,3,1,1,1,6,1,1,102,1,3,48,1,3,8,

1,1,20,1,1,1,3,1,1,784,1,1,3,1,1,1,20,1,1,17,84,1,1,3,

1,1,1,195,1,1,1,3,1,1,84,8,1,3,48,1,3,12,784,1,1,3,1,1,

1,20,1,1,17,84,1,1,3,1,1,1,195,1,1,1,3,1,1,84,8,1,3,48,

1,3,4198,84,1,1,3,6,1,3,1,1,1,6,1,1,10,28,1,1,3,323,1,1,

16463,1,1,1,3,1,1,84,1,1,3,1,1,1,6,1,1,34,1,1,6,1,1,1,

3,1,251,3,1,1,84,1,1,3,1,1,1,6,1,1,34,1,1,6,1,1,1,3,

1,13,3,1,4,1,1,6,1,1,83,1,1,20,1,1,1,3,1,1,784,8,1,3,

5,28,145,3,1,4,1,1,6,1,1,1,3,1,146,3,1,23,1,3,48,1,3,1,

1,1,6,1,1,4,1,3,59,3,1,1,84,1,1,3,1,1,1,6,1,1,10,28,

1,1,3,1,1,1,20,1,1,1,3,1,6,3,1,1,28,10,1,1,6,1,1,1,

3,1,1,84,1,1,3,2399,84,1,1,3,1,1,1,195,1,1,8,3,1,4,1,1,

6,1,1,1,3,1,48,3,1,1,28,5,3,1,58,1,3,1,1,1,20,1,1, ...].

...

GENERATING METHOD.

Start with CF = [1] and repeat (PARI code):

t = (1/4)*contfracpnqn(7*CF)[1,1]/contfracpnqn(7*CF)[2,1]; CF = contfrac(t)

This method can be illustrated as follows.

t0 = [1] = 1;

t1 = (1/4)*[7] = [1; 1, 3] = 7/4;

t2 = (1/4)*[7; 7, 21] = [1; 1, 3, 1, 1, 1, 20, 2] = 1057/592

t3 = (1/4)*[7; 7, 21, 7, 7, 7, 140, 14] = [1; 1, 3, 1, 1, 1, 20, 1, 1, 1, 3, 1, 1, 28, 35, 56] = 745628689/417608128;

t4 = (1/4)*[7; 7, 21, 7, 7, 7, 140, 7, 7, 7, 21, 7, 7, 196, 245, 392] = [1; 1, 3, 1, 1, 1, 20, 1, 1, 1, 3, 1, 1, 28, 35, 28, 1, 1, 3, 1, 1, 1, 20, 1, 1, 1, 3, 1, 1, 784, 61, 3, 1, 97, 4] = 381285245640002405521/213548405546586390784; ...

where this constant t equals the limit of the iterations of the above process.

PROG

(PARI) /* Generate over 5000 digits in the decimal expansion */

CF=[1];

{for(i=1, 10, t = (1/4)*contfracpnqn(7*CF)[1, 1]/contfracpnqn(7*CF)[2, 1];

CF = contfrac(t) ); }

for(n=1, 150, print1(floor(10^(n-1)*t)%10, ", "))

CROSSREFS

Cf. A320412, A320411, A320953.

Sequence in context: A216546 A003881 A225404 * A076419 A217516 A071420

Adjacent sequences: A320410 A320411 A320412 * A320414 A320415 A320416

KEYWORD

nonn,cons

AUTHOR

Paul D. Hanna, Oct 26 2018

STATUS

approved