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%I #35 Sep 08 2022 08:45:07
%S 1,12,287,6876,164737,3946812,94558751,2265463212,54276558337,
%T 1300371936876,31154649926687,746411226303612,17882714781360001,
%U 428438743526336412,10264647129850713887,245923092372890796876
%N Chebyshev sequence T(n,12) with Diophantine property.
%C a(143+286k)-1 and a(143+286k)+1 are consecutive odd powerful numbers. See A076445. - _T. D. Noe_, May 04 2006
%C Except for the first term, positive values of x (or y) satisfying x^2 - 24xy + y^2 + 143 = 0. - _Colin Barker_, Feb 19 2014
%H Vincenzo Librandi, <a href="/A077424/b077424.txt">Table of n, a(n) for n = 0..200</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 (24,-1).
%F a(n+1)^2 - 143*b(n)^2 = 1 for n>=0, with the companion sequence b(n)=A077423(n).
%F a(n) = 24*a(n-1) - a(n-2) for n>0, a(-1) := 12, a(0)=1.
%F a(n) = T(n, 12)= (S(n, 24)-S(n-2, 24))/2 = S(n, 24)-11*S(n-1, 24) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 24)=A077423(n).
%F a(n) = (ap^n + am^n)/2, with ap := 12+sqrt(143) and am := 12-sqrt(143).
%F a(n) = sum( ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*12)^(n-2*k), k=0..floor(n/2) ) for n>=1.
%F a(n+1) = sqrt(1 + 143*A077423(n)^2) for n>=0.
%F G.f.: (1-12*x)/(1-24*x+x^2).
%t CoefficientList[Series[(1 - 12 x)/(1 - 24 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 21 2014 *)
%o (Sage) [lucas_number2(n,24,1)/2 for n in range(20)] # _Zerinvary Lajos_, Jun 26 2008
%o (PARI) Vec((1-12*x)/(1-24*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 19 2014
%o (Magma) I:=[1,12]; [n le 2 select I[n] else 24*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 21 2014
%Y Cf. A090732.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 29 2002
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