Number of triads, continued

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

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