Hi Dick, No, sorry, I don't have any insights into the limit case. The OEIS page has a link to a paper that might help. Tom Richard Hess writes:
Thanks, Tom. Any thoughts on the second part of my post?
Sent from my iPhone
On Feb 13, 2020, at 8:03 AM, Tom Karzes <karzes@sonic.net> wrote:
I get the following for n = 0 through 10, with the results reduced to lowest-terms fractions:
f(0) = 0 f(1) = 2 f(2) = 8/3 f(3) = 22/7 f(4) = 368/105 f(5) = 2470/651 f(6) = 7880/1953 f(7) = 150266/35433 f(8) = 13315424/3011805 f(9) = 2350261538/513010785 f(10) = 1777792792/376207909
These values match yours if you reduce your fractions to lowest terms.
The numerators correspond to OEIS entry A158466. I don't think there's an entry for the non-lowest-terms numerators since those values are abitrary artifacts of the specific computation that was used to obtain the results.
Tom
Richard writes:
Dear Funsters Imagine n pennies in a frying pan. They are all flipped randomly at once and those landing heads are removed and placed heads up on a flat surface. The process is repeated with the remaining pennies until all coins show heads. What is the expected number of flips, f(n), for n coins? f(1) = 2, f(2) = 8/3, f(3) = 66/3/7, f(4) = 1104/3/7/15, f(5) = 37050/3/7/15/31 etc. The numerators follow the sequence 2, 8, 66, 1104, 37050, 2482200, 31336530, 88081529760, 46640940221610....
1. I don't find this sequence in the OEIS. Should it be there or is there a related sequence that is there?
2 f(64) = 7.34399... f(128) = 8.33837... f(256) = 9.33556... f(512) = 10.33415... f(1024) = 11.33345... f(2048) = 12.33309... f(4096) = 13.33292... f(8192) = 14.33283... f(16384) = 15.3327... Perhaps someone can verify these results and determine the limit of e(k) as k approaches infinity for f(2^k) = k+1+e(k). Any help is appreciated. Dick Hess