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%I #52 Sep 08 2022 08:45:14
%S 1,29,840,24331,704759,20413680,591291961,17127053189,496093250520,
%T 14369577211891,416221645894319,12056058153723360,349209464812083121,
%U 10115018421396687149,292986324755691844200,8486488399493666794651
%N Chebyshev polynomials S(n,29) with Diophantine property.
%C All positive integer solutions of Pell equation b(n)^2 - 837*a(n)^2 = +4 together with b(n)=A090251(n+1), n >= 0. Note that D=837=93*3^2 is not squarefree.
%C For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 29'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 For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,28}. - _Milan Janjic_, Jan 26 2015
%H Indranil Ghosh, <a href="/A097782/b097782.txt">Table of n, a(n) for n = 0..682</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (29, -1).
%F a(n) = S(n, 29) = U(n, 29/2) = S(2*n+1, sqrt(31))/sqrt(31) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
%F a(n) = 29*a(n-1) - a(n-2), n >= 1; a(0)=1, a(1)=29; a(-1)=0.
%F a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (29+3*sqrt(93))/2 and am = (29-3*sqrt(93))/2.
%F G.f.: 1/(1-29*x+x^2).
%F a(n) = Sum_{k=0..n} A101950(n,k)*28^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 0} (1 + 1/a(n)) = 1/9*(9 + sqrt(93)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 1} (1 - 1/a(n)) = 3/58*(9 + sqrt(93)). - _Peter Bala_, Dec 23 2012
%e (x,y) = (29;1), (839;29), (24302,840), ..., give the positive integer solutions to x^2 - 93*(3*y)^2 =+4.
%t LinearRecurrence[{29,-1},{1,29},20] (* _Harvey P. Dale_, Dec 14 2011 *)
%o (Sage) [lucas_number1(n,29,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 27 2008
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-29*x+x^2)) \\ _G. C. Greubel_, May 25 2019
%o (Magma) I:=[1,29]; [n le 2 select I[n] else 29*Self(n-1)-Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 25 2019
%o (GAP) a:=[1,29];; for n in [3..20] do a[n]:=29*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, May 25 2019
%Y Cf. A097781.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
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