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

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A090458 Decimal expansion of (3+sqrt 21)/2. 14
3, 7, 9, 1, 2, 8, 7, 8, 4, 7, 4, 7, 7, 9, 2, 0, 0, 0, 3, 2, 9, 4, 0, 2, 3, 5, 9, 6, 8, 6, 4, 0, 0, 4, 2, 4, 4, 4, 9, 2, 2, 2, 8, 2, 8, 8, 3, 8, 3, 9, 8, 5, 9, 5, 1, 3, 0, 3, 6, 2, 1, 0, 6, 1, 9, 5, 3, 4, 3, 4, 2, 1, 2, 7, 7, 3, 8, 8, 5, 4, 4, 3, 3, 0, 2, 1, 8, 0, 7, 7, 9, 7, 4, 6, 7, 2, 2, 5, 1, 6, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...

n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...

x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1...].  (See the Mathematica program below.) [From Clark Kimberling, May 05 2011]

Equals n+n/(n+n/(n+n/(n+....))) for n = 3.  See also A090388. - Stanislav Sykora, Jan 23 2014

REFERENCES

S. Sykora, Blazys Expansions and Continued Fractions, Stan's Library, Volume IV, Mathematics, 2013; http://www.ebyte.it/stan/2013_BlazysExpansions.pdf

LINKS

Table of n, a(n) for n=1..101.

EXAMPLE

3.79128784747792...

MATHEMATICA

FromContinuedFraction[{3, 1, {3, 1}}]

ContinuedFraction[%, 20]

RealDigits[N[%%, 120]] (*A090458*)

N[%%%, 40]

PROG

(PARI) solve(x=3, 4, x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(PARI) (3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

CROSSREFS

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).

Sequence in context: A118622 A101366 A217359 * A131712 A072845 A197481

Adjacent sequences:  A090455 A090456 A090457 * A090459 A090460 A090461

KEYWORD

easy,nonn,cons

AUTHOR

Felix Tubiana (fat2(AT)columbia.edu), Feb 05 2004

EXTENSIONS

Additional comments from Rick L. Shepherd, Jul 02 2004

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified April 23 12:19 EDT 2014. Contains 240920 sequences.