# [seqfan] Re: Higher dimensional analogues to plane division by ellipses?

Jess Tauber yahganlang at gmail.com
Sun Jun 2 20:49:37 CEST 2013

The shallow diagonals in the (2,1)-sided generalized Pascal Triangle sum to
Lucas numbers. Each of the terms in the shallow diagonals are also
identical to the numerical coefficients N of power terms NM^X in expansions
of equations defining the powers of the Metallic Means. The dimensional
labels of the side-parallel diagonals give X.

I still want to know if this is a known fact. I've corresponded with de
to the query.

[image: \!\ S_m^3 = S_{(m^3 + 3m)}]
[image: \!\ S_m^5 = S_{(m^5 + 5m^3 + 5m)}]
[image: \!\ S_m^7 = S_{(m^7 + 7m^5 + 14m^3 + 7m)}]
[image: \!\ S_m^9 = S_{(m^9 + 9m^7 + 27m^5 + 30m^3 + 9m)}]
[image: \!\ S_m^{11} = S_{(m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m)}.]

See: http://en.wikipedia.org/wiki/Silver_ratio

On Sun, Jun 2, 2013 at 1:55 PM, Hans Havermann <gladhobo at teksavvy.com>wrote:

> Yes, thank you. Spinadel's 1999 'The Family of Metallic Means' <
> http://www.mi.sanu.ac.rs/vismath/spinadel/ > with its bibliography of 30
> references, interestingly has a 1997 precursor (The Family of Metallic
> Means in Design) with *additional* references, including:
>
> [28] Godfrey Gumbs and M. K. Ali, Dynamical Maps, Cantor Spectra, and
> Localization for Fibonacci and Related Quasiperiodic Lattices, Phys. Rev.
> Lett., vol. 60, 1988.
>
> [29] Godfrey Gumbs and M. K. Ali, Quasiperiodic dynamics for a generalized
> third-order Fibonacci series, Physical Review B, vol. 38, Nr. 10, october
> 1988.
>
> [30] Godfrey Gumbs and M. K. Ali , Electronic Properties of the
> Tight-Binding Fibonacci Hamiltonian, J. Phys. A: Math. Gen. vol. 22, 1989.
>
> [31] Kolár M. and M. K. Ali, Generalized Fibonacci superlattices,
> dynamical trace maps, and magnetic excitations, Phys. Rev. B, vol. 39, Nr.
> 1, 1989.
>
> [32] Kolár M. and M. K. Ali, Attractors in quantum Ising models, Phys.
> Rev. B, vol. 40, Nr. 16, 1989.
>
> Reference 31 < http://prb.aps.org/abstract/PRB/v39/i1/p426_1 > mentions
> "spin-wave spectra for Fibonacci superlattices with copper and nickel mean
> are compared with those for the golden-mean case".
>
>
> On Jun 2, 2013, at 12:54 PM, allouche at math.jussieu.fr wrote:
>
> > It is that clear that the expression "metallic mean"
> > was coined in 1999?
> >
> > I seem to remember having heard the expression
> > "metallic mean" before (i.e., some years after
> > the discovery of quasicrystals).
> >
> > In particular I traced back a 1996 paper:
> > J. A. G. Roberts, Escaping orbits in trace maps,
> > Physica A: Statistical Mechanics and its Applications,
> > Volume 228, Issues 1?4, 15 June 1996, Pages 295--325
> > where the term "metallic mean" is used (see Page 298
> > just before (11)). In that paper "metallic-mean" sequences
> > are generated by the substitution rule (morphism of the
> > free monoid) a --> b, b --> b^{\ell} a, whose transition
> > matrix admits (1 + \sqrt{1+4\ell})/2 as dominant eigenvalue.
> >
> > Other "metals" were used before, e.g., bronze in several
> > papers including
> > G. Gumbs and M. K. Ali, Scaling and eigenstates for a class of
> > one-dimensional quasiperiodic lattices, J. Phys. A: Math. Gen.
> > 1988, 21 L517--L521.
> >
> > I am not even sure that Roberts' paper is the first one where
> > the expression is used
>
>
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