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%I #39 Sep 08 2022 08:45:07
%S 1,11,241,5291,116161,2550251,55989361,1229215691,26986755841,
%T 592479412811,13007560326001,285573847759211,6269617090376641,
%U 137646002140526891,3021942430001214961,66345087457886202251,1456569981643495234561,31978194508699008958091
%N Chebyshev sequence T(n,11) with Diophantine property.
%C Numbers n such that 30*(n^2-1) is square. - _Vincenzo Librandi_, Aug 08 2010
%C Except for the first term, positive values of x (or y) satisfying x^2 - 22xy + y^2 + 120 = 0. - _Colin Barker_, Feb 19 2014
%H Colin Barker, <a href="/A077422/b077422.txt">Table of n, a(n) for n = 0..745</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 (22,-1).
%F a(n+1)^2 - 30*(2*b(n))^2 = 1, n>=0, with the companion sequence b(n)=A077421(n).
%F a(n) = 22*a(n-1) - a(n-2), a(-1) := 11, a(0)=1.
%F a(n) = T(n, 11) = (S(n, 22)-S(n-2, 22))/2 = S(n, 22)-11*S(n-1, 22) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 22)=A077421(n).
%F a(n) = (ap^n + am^n)/2 with ap := 11+2*sqrt(30) and am := 11-2*sqrt(30).
%F a(n) = sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*11)^(n-2*k), k=0..floor(n/2)), n>=1.
%F a(n+1) = sqrt(1 + 30*(2*A077421(n))^2), n>=0.
%F a(n) = Cosh[2n*ArcSinh[Sqrt[5]]] - _Herbert Kociemba_, Apr 24 2008
%F G.f.: (1-11*x)/(1-22*x+x^2). - _Philippe Deléham_, Nov 17 2008
%t Table[Cos[n*ArcCos[11]] // Round, {n, 0, 15}] (* _Jean-François Alcover_, Dec 19 2013 *)
%t LinearRecurrence[{22,-1},{1,11},20] (* _Harvey P. Dale_, Jul 30 2022 *)
%o (Sage) [lucas_number2(n,22,1)/2 for n in range(0,20)] # _Zerinvary Lajos_, Jun 26 2008
%o (Magma) [n: n in [1..10000000] |IsSquare(30*(n^2-1))] // _Vincenzo Librandi_, Aug 08 2010
%o (PARI) Vec((1-11*x)/(1-22*x+x^2) + O(x^100)) \\ _Colin Barker_, Jun 15 2015
%Y Cf. A090730.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 29 2002