source file: mills2.txt Date: Thu, 25 Jul 1996 07:42:54 -0700 Subject: Number of triads, continued From: COUL@ezh.nl (Manuel Op de Coul) The integer series I gave on 22 July for the number of different triads in equal temperaments have been identified by Neil Sloane of AT&T who is an integer series specialist. See his webpages: http://netlib.att.com/netlib/att/math/sloane/doc/eistop.html 1 1 2 3 4 5 7 8 10 12 14 16 19 21 24 27 30 33 37 40 44 48 52 56 61 65 catalogue number: A001399 Partitions into at most 3 parts. The generating function is 1 - ----------------------------- 2 3 (x + 1) (x + x + 1) (x - 1) 1 1 2 4 5 7 10 12 15 19 22 26 31 35 40 46 51 57 64 70 77 85 92 100 109 catalogue number: A007997 Solutions to x+y+z=0 (mod n). Molien series for A sub 3. D.J. Benson, Poly. Invts. of Finite Grps, Cambr., 1993, p. 105. Generating function: 2 (x - x + 1) - --------------------- 2 3 (x + x + 1) (x - 1) To know more about generating functions one can take a book about combinatorial mathematics. They are not expressions that produce the series however. With some manipulations, i.e. splitting them in separate terms and then doing an inverse transformation, it might be possible to obtain a closed form expression for the series, with binomial coefficients probably. So whoever wants to take a shot, go ahead. Manuel Op de Coul coul@ezh.nl Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 25 Jul 1996 18:35 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA10866; Thu, 25 Jul 1996 09:35:21 -0700 Date: Thu, 25 Jul 1996 09:35:21 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu