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!)
A181784 Numerators of a series sum related to a game of chance. 1
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3782992, 32389076, 275617830, 2350749914, 20140518790, 173429992350, 1500850805160, 14550277251918, 133009333771170, 1198324107797254 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Consider a 1-dimensional random walk from 0 with equal-probability steps of Pi and -1. One way to compute the probability of eventually walking below 0 is as the sum over n of the probabilities of becoming negative after a walk with exactly n steps of Pi (n >= 0) and max(ceiling(n*Pi),1) steps of -1. The total number of walks of such length for a given n is 2^(n+max(ceiling(n*Pi),1)), or 2^(n+A004084(n)) (n >= 1), forming a sequence of denominators, and this sequence gives the numerators, the number of possible sequences of length (n+max(ceiling(n*Pi),1)) drawn from {Pi, -1} such that no partial sum except the total sum is < 0.

See the Munafo web page for complete description.

a(n) diverges from A002293 because Pi is not exactly 3.

LINKS

Table of n, a(n) for n=0..18.

R. Munafo, Related to a Game of Chance

"My Math Forum" discussion thread, I give, duz... what is it?

"duz" blog entry, Random Walking

EXAMPLE

Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...

CROSSREFS

Sequence in context: A216712 A240586 A002293 * A003287 A077056 A227404

Adjacent sequences: A181781 A181782 A181783 * A181785 A181786 A181787

KEYWORD

nonn,frac

AUTHOR

Robert Munafo, Dec 21 2010

EXTENSIONS

a(18) from Robert Munafo, Dec 22 2010

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 December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)