login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144229 The numerators of the convergents to the recursion x=1/(x^2+1). 1
1, 1, 4, 25, 1681, 5317636, 66314914699609, 8947678119828215014722891025, 178098260698995011212395018312912894502905113202338936836 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The recursion converges to the real root of 1/(x^2+1) - x = 0, 0.682327803...

An interesting consequence of this result occurs if we multiply by x^2+1 to get 1-x-x^3=0. These different equations intersect at the same root 0.682327803... Note also that a(n) is a square. The square roots form sequence A076725.

a(n) is the number of (0,1)-labelled perfect binary trees of height n such that no adjacent nodes have 1 as the label and the root is labelled 0. - Ran Pan, May 22 2015

LINKS

Table of n, a(n) for n=0..8.

Ran Pan, Exercise R, Project P.

FORMULA

a(n+2) = (a(n)^2 + a(n+1))^2. - Ran Pan, May 22 2015

a(n) ~ c * d^(2^n), where c = A088559 = 0.465571231876768... is the root of the equation c*(1+c)^2 = 1, d = 1.6634583970724267140029... . - Vaclav Kotesovec, May 22 2015

MATHEMATICA

f[n_]:=(n+1/n)/n; Prepend[Denominator[NestList[f, 2, 7]], 1] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

RecurrenceTable[{a[n]==(a[n-2]^2 + a[n-1])^2, a[0]==1, a[1]==1}, a, {n, 0, 10}] (* Vaclav Kotesovec, May 22 2015 after Ran Pan *)

PROG

(PARI) x=0; for(j=1, 10, x=1/(x^2+1); print1((numerator(x))", "))

CROSSREFS

Cf. A076725, A088559.

Sequence in context: A072882 A014253 A132553 * A284106 A063802 A276266

Adjacent sequences:  A144226 A144227 A144228 * A144230 A144231 A144232

KEYWORD

frac,nonn

AUTHOR

Cino Hilliard, Sep 15 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 23 21:46 EST 2017. Contains 295141 sequences.