Comments on: The oddest numbers http://bit-player.org/2006/the-oddest-numbers An amateur's outlook on computation and mathematics. Fri, 12 Mar 2010 05:42:09 +0000 http://wordpress.org/?v=2.6.3 By: Rainer Rosenthal http://bit-player.org/2006/the-oddest-numbers#comment-197 Rainer Rosenthal Sun, 04 Jun 2006 15:08:22 +0000 http://bit-player.org/?p=44#comment-197 I found your interesting website while searching for a connection between binary trees and the so-called Josephus problem. Standing in a circle and counting to 3, throwing out every third person in the circle and proceeding with the next one, how many people do you need at least for lasting the game n rounds? The number a(n) is - astonishingly - given by Sloanes http://www.research.att.com/~njas/sequences/A112088. I am not familiar with the concept of binary trees and don't really understand what this sequence is counting. I asked this in the newsgroup alt.math.recreational today. Maybe you like to join? Best regards, Rainer I found your interesting website while searching for a connection between binary trees and the so-called Josephus problem.

Standing in a circle and counting to 3, throwing out every third
person in the circle and proceeding with the next one, how many
people do you need at least for lasting the game n rounds?
The number a(n) is - astonishingly - given by Sloanes
http://www.research.att.com/~njas/sequences/A112088.

I am not familiar with the concept of binary trees and don’t really understand what this sequence is counting.
I asked this in the newsgroup alt.math.recreational today.
Maybe you like to join?

Best regards,
Rainer

]]> By: Christian Duncan http://bit-player.org/2006/the-oddest-numbers#comment-120 Christian Duncan Sat, 13 May 2006 18:57:33 +0000 http://bit-player.org/?p=44#comment-120 Did you also notice that odds(n) = fusc(n) - 1? Like the previous comment, you can use induction to prove it. But there may be a simpler visual proof for you. Look at the tree that you created for the recursive calls. If you had an even number, you had a single branch ("delete/compress these"). The odd numbers (>1) give you a double branch. What is the result? A true binary tree. Each internal node in this tree corresponds to one odd call and the leaf nodes correspond to terminal calls (n=1). And, we know that the number of internal nodes of a binary tree is exactly one less than the number of external (leaf) nodes. Did you also notice that odds(n) = fusc(n) - 1?

Like the previous comment, you can use induction to prove it. But there may be a simpler visual proof for you. Look at the tree that you created for the recursive calls. If you had an even number, you had a single branch (”delete/compress these”). The odd numbers (>1) give you a double branch. What is the result? A true binary tree. Each internal node in this tree corresponds to one odd call and the leaf nodes correspond to terminal calls (n=1).

And, we know that the number of internal nodes of a binary tree is exactly one less than the number of external (leaf) nodes.

]]> By: Jonathan Katz http://bit-player.org/2006/the-oddest-numbers#comment-118 Jonathan Katz Fri, 12 May 2006 11:04:49 +0000 http://bit-player.org/?p=44#comment-118 Actually, a simple proof by induction shows that evens(n) is greater than or equal to odds(n). The only "trick" is that you need to prove the following stronger claim: if n is even then evens(n) is strictly greater than odds(n), while if n is odd then evens(n) is greater than or equal to odds(n). Actually, a simple proof by induction shows that evens(n) is greater than or equal to odds(n). The only “trick” is that you need to prove the following stronger claim: if n is even then evens(n) is strictly greater than odds(n), while if n is odd then evens(n) is greater than or equal to odds(n).

]]> By: Jess Austin http://bit-player.org/2006/the-oddest-numbers#comment-114 Jess Austin Thu, 11 May 2006 01:51:00 +0000 http://bit-player.org/?p=44#comment-114 You might be interested in "Enumerating the Rationals", by Gibbons, Lester, and Bird. This paper may be found at http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/rationals.pdf You might be interested in “Enumerating the Rationals”, by Gibbons, Lester, and Bird. This paper may be found at http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/rationals.pdf

]]>