|
EXAMPLE
|
The decimal expansion of this constant t begins:
t = 1.540546562581850919585092443277493630777443307008831680830472910...
The simple continued fraction expansion of t begins:
t = [1; 1, 1, 5, 1, 1, 1, 120, 1, 1, 1, 5, 1, 1, 1, 2880, 1, 1, 1, 5, 1, 1, 1, 120, 1, 1, 1, 5, 1, 1, 1, 69120, 1, 1, 1, 5, ... , a(n), ...]
such that the simple continued fraction expansion of 4*t begins:
4*t = [6; 6, 6, 30, 6, 6, 6, 720, 6, 6, 6, 30, 6, 6, 6, 17280, 6, 6, 6, 30, 6, 6, 6, 720, 6, 6, 6, 30, 6, 6, 6, 414720, 6, 6, 6, 30, ... , 6*a(n), ...].
The initial 528 terms in the continued fraction expansion of t are
t = [1;1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,69120,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,1658880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,69120,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,39813120,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,69120,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,1658880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,69120,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,955514880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,69120,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,1658880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,69120,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,39813120,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,69120,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,1658880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,69120,
1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880,1,1,1,5,1,1,1,120,
1,1,1,5,1,1,1,22932357120,1,1,1,5,1,1,1,120,1,1,1,5,1,1,1,2880, ...].
...
The initial 1000 digits in the decimal expansion of t are
t = 1.54054656258185091958509244327749363077744330700883\
16808304729109136071030553148439916006575514781913\
07822865753878621983372629507214585549765601216855\
82787499037533808802512182117029618023317430110304\
68568548056881176292575460160397807605484819864109\
97669631806987876538679522272871069799153383954670\
27608251838807810531236786249017655750340240450431\
39500820756134042709955133598762098341500330573137\
99939807838257483585300963279557471021988938954240\
72201092140208383999296983006564129234149599645137\
97256177580945556498719774507283674158287467111752\
77844466938413199720373428083674374904367749851098\
21424896071080046214454673955084637450020810561194\
71345165789766421139563276393063288950816919055445\
03342967346596573587017045564281666931351950277058\
62590013151270263215075119134200880720738116296046\
68850174252499836432241162269740091282255326351721\
57235179003932955261950046664886398112095558223840\
82192509104134957717517226738372957214588632890700\
21161549012492823004735717009953284171956638331978...
...
GENERATING METHOD.
Start with CF = [1] and repeat (PARI code):
t = (1/4)*contfracpnqn(6*CF)[1,1]/contfracpnqn(6*CF)[2,1]; CF = contfrac(t)
This method can be illustrated as follows.
t0 = [1] = 1 ;
t1 = (1/4)*[6] = [1; 2] = 3/2 ;
t2 = (1/4)*[6; 12] = [1; 1, 1, 11, 2] = 73/48 ;
t3 = (1/4)*[6; 6, 6, 66, 12] = [1; 1, 1, 5, 1, 1, 1, 264, 3] = 45312/29413 ;
t4 = (1/4)*[6; 6, 6, 30, 6, 6, 6, 1584, 18] = [1; 1, 1, 5, 1, 1, 1, 120, 1, 1, 1, 5, 1, 1, 1, 6336, 4, 2] = 22476134901/14589714746 ;
t5 = (1/4)*[6; 6, 6, 30, 6, 6, 6, 720, 6, 6, 6, 30, 6, 6, 6, 38016, 24, 12] = [1; 1, 1, 5, 1, 1, 1, 120, 1, 1, 1, 5, 1, 1, 1, 2880, 1, 1, 1, 5, 1, 1, 1, 120, 1, 1, 1, 5, 1, 1, 1, 152064, 6, 48] = 19666327045008787868161/12765811513057655118144 ; ...
where this constant t equals the limit of the iterations of the above process.
The number of terms generated in each iteration of the above method begins:
[1, 2, 5, 9, 18, 34, 66, 133, 265, 530, 1061, 2122, 4242, 8485, 16969, 33938, ...].
...
GENERATING FUNCTION.
The terms of this continued fraction may be expressed by the generating function
A(x) = 1 + x + x^2 + 5*x^3 + x^4 + x^5 + x^6 + 120*x^7 + x^8 + x^9 + x^10 + 5*x^11 + x^12 + x^13 + x^14 + 2880*x^15 + ...
where A(x) = 1/(1-x) + 4*x^3/(1-x^4) + 115*x^7/(1-x^8) + 2760*x^15/(1-x^16) + 66240*x^31/(1-x^32) + 1589760*x^63/(1-x^64) + 38154240*x^127/(1-x^128) + ... + 115*24^(n-3)*x^(2^n-1)/(1 - x^(2^n)) + ...
...
|