Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site utcsstat.UUCP Path: utzoo!utcsstat!davids From: davids@utcsstat.UUCP (David Scollnik) Newsgroups: net.math Subject: Pascal's Inverse Triangle Message-ID: <2216@utcsstat.UUCP> Date: Wed, 3-Jul-85 12:55:32 EDT Article-I.D.: utcsstat.2216 Posted: Wed Jul 3 12:55:32 1985 Date-Received: Wed, 3-Jul-85 13:52:43 EDT Organization: U. of Toronto, Canada Lines: 70 Undoubtedly, the great majority of you are familiar with Pascal's triangle, that is, this triangle which continues off to infinity ... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 note, every entry in the triangle is the sum of the two entries directly above. Now, consider this, which I shall refer to as Pascal's Inverse triangle... 1 1 1 1 1/2 1 1 2/3 2/3 1 1 3/5 3/4 3/5 1 this triangle also continues on for infinity. Each entry is the INVERSE of the sum of the two entries directly above, for example 1/2 = 1/(1 + 1) and 2/3 = 1/(1/2 + 1) and 3/4 = 1/(2/3 + 2/3) . Of course, the outermost layer will always be one, and it can be shown (for example with the use of continued fractions) that the next layer inward will converge to (sqrt(5)-1)/2. This is apparent since we have, as the number of levels in the triangle goes off to infinity, that 1 X = ----------------- 1 + 1 ----------- 1 + 1 --------- 1 + etc. and this is the same as 1 X = --------------- 1 + X X * X + X - 1 = 0 and this equations one positive root is (sqrt(5)-1)/2 Has anyone come across this triangle before, and if so have you anything of interest to share concerning it ?