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A097781 Chebyshev polynomials S(n,27) with Diophantine property.

%I A097781 #48 Sep 08 2022 08:45:14
%S A097781 1,27,728,19629,529255,14270256,384767657,10374456483,279725557384,
%T A097781 7542215592885,203360095450511,5483180361570912,147842509666964113,
%U A097781 3986264580646460139,107481301167787459640,2898008866949614950141
%N A097781 Chebyshev polynomials S(n,27) with Diophantine property.
%C A097781 All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree.
%C A097781 For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 27's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C A097781 For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,26}. - _Milan Janjic_, Jan 26 2015
%H A097781 Vincenzo Librandi, <a href="/A097781/b097781.txt">Table of n, a(n) for n = 0..700</a>
%H A097781 R. Flórez, R. A. Higuita, A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H A097781 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097781 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A097781 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (27,-1).
%F A097781 a(n) = S(n, 27) = U(n, 27/2) = S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
%F A097781 a(n) = 27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.
%F A097781 a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (27+5*sqrt(29))/2 and am = (27-5*sqrt(29))/2.
%F A097781 G.f.: 1/(1-27*x+x^2).
%F A097781 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - _Philippe Deléham_, Feb 10 2012
%F A097781 Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(29)). - _Peter Bala_, Dec 23 2012
%F A097781 Product {n >= 1} (1 - 1/a(n)) = 5/54*(5 + sqrt(29)). - _Peter Bala_, Dec 23 2012
%e A097781 (x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.
%p A097781 with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); # _Zerinvary Lajos_, Apr 20 2008
%t A097781 Join[{a=1,b=27},Table[c=27*b-a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2011 *)
%t A097781 CoefficientList[Series[1/(1 - 27 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%o A097781 (Sage) [lucas_number1(n,27,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o A097781 (Magma) I:=[1, 27, 728]; [n le 3 select I[n] else 27*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 24 2012
%Y A097781 Cf. A078362, A078366.
%K A097781 nonn,easy
%O A097781 0,2
%A A097781 _Wolfdieter Lang_, Aug 31 2004