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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 (list; graph; refs; listen; history; text; internal format)
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
Sequence in context: A307086 A021090 A368669 * A177889 A086744 A242301
KEYWORD
nonn,cofr
AUTHOR
Andrew Hone, Mar 11 2016
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)