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%I #44 Sep 08 2022 08:45:14
%S 1,13,337,8749,227137,5896813,153090001,3974443213,103182433537,
%T 2678768828749,69544807113937,1805486216133613,46873096812360001,
%U 1216895030905226413,31592397706723526737,820185445343906468749,21293229181234844660737,552803773266762054710413
%N Chebyshev T-polynomials T(n,13) with Diophantine property.
%C a(n)^2 - 42 (2*b(n))^2 = +1 with b(n):=A097309(n) gives all nonnegative solutions of this D:= 42*4 = 168 Pell equation.
%C Numbers n such that 42*(n^2-1) is a square. - _Vincenzo Librandi_, Nov 17 2010
%C Except for the first term, positive values of x (or y) satisfying x^2 - 26xy + y^2 + 168 = 0. - _Colin Barker_, Feb 20 2014
%H Seiichi Manyama, <a href="/A097308/b097308.txt">Table of n, a(n) for n = 0..700</a> (terms 0..200 from Vincenzo Librandi)
%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 (26,-1).
%F a(n) = 26*a(n-1) - a(n-2), a(-1) := 13, a(0)=1.
%F a(n) = T(n, 13) = (S(n, 26) - S(n-2, 26))/2 = S(n, 26) - 13*S(n-1, 26) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 26)=A097309(n).
%F a(n) = (ap^n + am^n)/2 with ap := 13+2*sqrt(42) and am := 13-2*sqrt(42).
%F a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*13)^(n-2*k), n >= 1.
%F G.f.: (1 - 13*x)/(1 - 26*x + x^2).
%F a(n) = sqrt(1 + 168*A097309(n)^2), n >= 0.
%F a(n) = cosh(2n*arcsinh(sqrt(6))). - _Herbert Kociemba_, Apr 24 2008
%t CoefficientList[Series[(1 - 13 x)/(1 - 26 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 22 2014 *)
%t LinearRecurrence[{26,-1},{1,13},20] (* _Harvey P. Dale_, Jul 01 2019 *)
%o (PARI) Vec((1-13*x)/(1-26*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 20 2014
%o (Magma) I:=[1,13]; [n le 2 select I[n] else 26*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 22 2014
%Y Cf. A097309.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 31 2004
%E Additional terms from _Colin Barker_, Feb 20 2014
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