Hi mathfun - Lately, a student, JB, and I have been looking at the ranks of the numbers j*x when the numbers i+j*x, for any positive real x, are arranged in order. These rank sequences are closely related to signature sequences. If x is rational, then there are repeated terms, so we extend the notion of rank to "minrank" and "maxrank". For example, if you order the numbers i+j*2 for i>=0, j>=0, then minrank(2n) means the rank of the first appearance of 2n and maxrank(2n) means the rank of the last appearance of 2n. For x=2, maxrank(2n) gives squares; more generally, i*mj for integral m gives figurate numbers for maxrank and central polygonal numbers for minrank. Returning to i+j*x for rational x, limits provide insights in case x is irrational, and Farey fractions play a central role. Linked to rank sequences are notions of 1st partial complement of a sequence, 2nd partial complement, and so on, with applications to dispersions of sequences. Loosely speaking, if A is the dispersion of the complement of a rank sequence r of x>1, then (1) r is column 1 of A; (2) the 1st partial complement of r is, after the 1st term, row 1 of A; (3) row 1 of A is the rank sequence of 1/x. If the sequence r in column 1 is strictly increasing (but not necessarily a rank sequence), let t(r) denote the sequence in row 1; we expect to prove that t is involutory if and only if r is a rank sequence, and that the sequence r, t(t(r)), t(t(t(t(r)))), ... converges to a rank sequence. A preprint should be ready soon. Now, JB and I need enlightenment on origins of terminology. Was Lancelot Hogben first to give names to the sequences which he called central polygonal numbers? The first three of these sequences are Central polygonal numbers, A002061: 1,3,7,13,21,... Central triangular numbers, A005448: 1,4,10,19,31,... Central square numbers, A001844: 1,5,13,25,41,... Next, as far as I know, the terms "signature sequence" and "fractal sequence" (for an infinitive sequence that contains itself as a proper subsequence) first appeared in 1995. It would seems likely that someone considered signature sequences (or rank sequences) before 1995, perhaps under some other name. Any insights about these matters? As a final thought, consider the ranks of the numbers 3^j in the ordered sequence of the numbers (2^i)*(3^j). (What's x?) Best regards, Clark Kimberling -----Original Message----- From: lkmitch@att.net [mailto:lkmitch@att.net] Sent: Wednesday, May 28, 2003 1:57 PM To: math-fun Subject: [math-fun] Signatures of irrational numbers Hi all, Lately, I've been playing with the signature of irrational numbers. My understanding, from what I've read on the web, is this: For a positive irrational number x, form the numbers y = i + j*x, where i and j are both positive integers. Since x is irrational, no 2 y values will be the same for different i and j. Arrange the ys by size, then the sequence of i values is a fractal sequence, and the signature of x. For x = phi ~ 1.618034, the first few entries are: i j y 1 1 2.618033989 2 1 3.618033989 1 2 4.236067977 3 1 4.618033989 2 2 5.236067977 4 1 5.618033989 1 3 5.854101966 3 2 6.236067977 5 1 6.618033989 And the signature begins: 1, 2, 1, 3, 2, 4, 1, 3, 5. If you strike the first occurence of every integer in the sequence, you get the original sequence back, which makes this a fractal sequence. My questions are: - What about the j sequence? From what I've seen experimentally, it seems to be a fractal sequence, too. Why is the signature the i sequence as opposed to the j sequence? What's known about the relation of the j sequence to the i sequence? - For a limited set of integers (both i and j run from 1 to 50), I plotted i vs. j, and the result was very interesting (I thought). You can find the picture here: http://www.fractalus.com/kerry/sigofphi.html The plot is one continuous zig-zag line which seems to never cross itself. But, the angle of the line changes slightly, causing some areas to bunch up and appear darker, and others to spread out and appear lighter. The overall effect is of a series of rectangles drawn in different shades of gray. Can anyone point me to other work that has been done on this? Thanks, Kerry Mitchell -- lkmitch@att.net www.fractalus.com/kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun