login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054765 a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 0, a(1) = 1. 8
0, 1, 3, 19, 160, 1744, 23184, 364176, 6598656, 135484416, 3108695040, 78831037440, 2189265960960, 66083318415360, 2154235544616960, 75425161203302400, 2822882994841190400, 112463980097804697600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The denominators of the convergents of [1/3, 4/5, 9/7, 16/9, ...]. To produce Pi the above continued fraction is used. It is formed by n^2/(2*n+1) which starts at n=1. Most numerators of continued fractions are 1 & thus are left out of the brackets. In the case of Pi they vary. Therefore here both numerators & denominators are given. The first 4 convergents are 1/3,5/19,44/160,476/1744. The value of this continued fraction is .273239... . 4*INV(1+.273239...) is Pi. - Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

Starting with offset 1 = row sums of triangle A155729. [Gary W. Adamson & Alexander R. Povolotsky, Jan 25 2009]

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..390

K. S. Brown, Integer Sequences Related To Pi

FORMULA

a(n) ~ Pi * (1+sqrt(2))^(n + 1/2) * n^n / (2^(9/4) * exp(n)). - Vaclav Kotesovec, Feb 18 2017

MAPLE

A054765 := proc(n)

option remember;

if n <= 1 then

n;

else

(2*n-1)*procname(n-1)+(n-1)^2*procname(n-2) ;

end if;

end proc: # R. J. Mathar, Jul 13 2013

MATHEMATICA

RecurrenceTable[{a[n + 2] == (2*n + 3)*a[n + 1] + (n + 1)^2*a[n],

a[0] == 0, a[1] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Feb 18 2017 *)

CROSSREFS

Cf. A155729, A012244, A054766.

Sequence in context: A307697 A320352 A301921 * A232691 A057719 A289258

Adjacent sequences: A054762 A054763 A054764 * A054766 A054767 A054768

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 26 2000

EXTENSIONS

More terms from James A. Sellers, May 27 2000

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)