login
A270138
Continued fraction expansion of the constant 6/A270121(1)+Sum_{n>=2}1/A270121(n).
0
0, 1, 6, 2, 7, 32, 112, 10800, 403200, 17418254400, 1755760043520000
OFFSET
0,3
COMMENTS
A270121 is defined by the following recurrence: if A270121(n)=x(n) then x(n+1)*x(n-1)=x(n)^2*(1+n*x(n)) for n>=1, with x(1)=7, x(2)=112; and for A270124, if A270124(n)=y(n) then y(0)=2 and y(n)=x(n+1)/x(n) for n>=1. Both of these sequences appear in this continued fraction expansion, which defines a transcendental number.
LINKS
A. N. W. Hone, Curious continued fractions, nonlinear recurrences and transcendental numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4.
A. N. W. Hone, Continued fractions for some transcendental numbers, arXiv:1509.05019 [math.NT], 2015-2016, Monatsh. Math. DOI: 10.1007/s00605-015-0844-2.
FORMULA
a(2*n+1) = n*A270124(n-1), a(2*n+2) = A270121(n) for n>=1.
EXAMPLE
6/A270121(1)+Sum_{n>=2}1/A270121(n)=6/7+1/112+1/403200+1/1755760043520000+...
=[0;1,6,2,7,32,112,10800,403200,17418254400,...]
=[0;1,6,A270124(0),A270121(1),2*A270124(1),A270121(2),3*A270124(2),A270121(3),4*A270124(3),...] (continued fraction).
CROSSREFS
KEYWORD
nonn,cofr
AUTHOR
Andrew Hone, Mar 11 2016
STATUS
approved