[seqfan] Re: A049310 (by way of moderator)

[Joerg Arndt]

> Here is the original message By L. Edson Jeffery:
> [...]

> 
> The so-called "Chebyshev S-polynomials" are defined by the recurrence
> 
> S(N,x) = x*S(N-1,x) - S(N-2,x)  (N>1),
> 
> with initial conditions S(0,x) = 1, S(1,x) = x. The coefficients for S(N,x)
> appear in row N of triangle http://oeis.org/A049310.
> 
> These polynomials appear to be periodic with period 2n when putting x =
> 2*cos(k*Pi/n), for any nonzero integer k and any integer n>1. If true, then
> the indices can be reduced modulo 2n, a property I have found to be quite
> useful for rhombus substitution tilings over the past thirty years.
> 
> [...]

If these polynomials are some simple transform of the Chebyshev
polynomials of the first kind, then that cos-formula surely holds.

About combinatorics and Chebyshev-polys I find (raw copy & paste):

%{Arthur T.\ Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck:
%{Combinatorial trigonometry with Chebyshev polynomials},
%Journal of Statistical Planning and Inference, vol.~140, no.~8, pp.~2157-2160, \bdate{August-2010}.
http://www.sciencedirect.com/science/article/pii/S0378375810000224

%{Arthur T.\ Benjamin, Daniel Walton:
%{Combinatorially composing Chebyshev polynomials},
%Journal of Statistical Planning and Inference, vol.~140, no.~8, pp.~2161-2167, \bdate{August-2010}.
%% http://www.sciencedirect.com/science/article/pii/S0378375810000236  Ohm-subscription
%% proof (via n-tilings) of
%% T_m(T_n(x))=T_mn(x)T_m(T_n(x))=T_mn(x) and U_m-1(T_n(x))U_n-1(x)=U_mn-1(x).

%{Qing-Hu Houa, Toufik Mansour:
%{Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials},
%Discrete Applied Mathematics, vol.~154, no.~8, pp.~1183-1197, \bdate{15-May-2006}.
%URL: \url{http://www.sciencedirect.com/science/article/pii/S0166218X05003586}.}
%% A071359

%{Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger:
%{Smooth words and Chebyshev polynomials},
%arXiv:0809.0551v1 [math.CO], \bdate{3-September-2008}.
%URL: \url{http://arxiv.org/abs/0809.0551v1}.}
%\jjfile{knopfmacher-smooth-words-chebyshev-polys.pdf}
%% seq. A215329 - A215334, A208772 - A208777 (necklace)
%% seq. A001333(3), A126358(4), A126359(5), A057960(6), A126361(7)=A002714, (words)
%% cf. http://www.jstor.org/stable/10.2307/3613451 for earlier reference

%{}{Toufik Mansour, Mark Shattuck:
%{Chebyshev Polynomials and Statistics on a New Collection of Words in the Catalan Family},
%arXiv:1407.3516 [math.CO], \bdate{14-July-2014},
%URL: \url{http://arxiv.org/abs/1407.3516}.}

Best regards,   jj