A114552 Odd-indexed bisection of A114551, which is the continued fraction expansion of Sum_{n>=0} 1/A112373(n); also, a(n) = A112373(n+1)/A112373(n) for n>=0.

1, 2, 6, 78, 73086, 4999703411742, 1710009514450915230711940280907486, 1000118217480414644596377710149364954622541926875237885435586055466747491994623223953758

Offset

0,2

Comments

A112373 is defined by the recurrence: let b(n) = A112373(n), then

b(n) =(b(n-1)^3 + b(n-1)^2)/b(n-2) for n>=2 with b(0)=b(1)=1.

The next term 58499...45086 has 228 digits and is too large to include.

Links

Table of n, a(n) for n=0..7.

Andrew N. W. Hone, Curious continued fractions, nonlinear recurrences and transcendental numbers, arXiv:1507.00063 [math.NT], 2015  and J. Int. Seq. 18 (2015) # 15.8.4.

Formula

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

a(n) = a(n-1)*(1 - a(n-1) + a(n-1)^2/a(n-2)) for n>=2, with a(0)=1, a(1)=2.

Example

a(5) = 4999703411742 = 73086*(1 + 1*2*6*78*73086).
a(5) = a(4)*(1 - a(4) + a(4)^2/a(3)) = 73086*(1 - 73086 + 73086^2/78).

Prog

(PARI) a(n)=if(n==0, 1, if(n==1, 2, a(n-1)*(1-a(n-1)+a(n-1)^2/a(n-2))))

Crossrefs

Cf. A112373, A114550 (constant), A114551 (continued fraction).

Sequence in context: A136306 A274825 A076146 * A302343 A244084 A362581

Adjacent sequences: A114549 A114550 A114551 * A114553 A114554 A114555

Keyword

nonn

Author

Paul D. Hanna, Dec 08 2005

Extensions

a(7) corrected by Georg Fischer, Aug 26 2022

Status

approved