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%I #24 Mar 06 2021 01:31:28
%S 2,51,2599,132498,6754799,344362251,17555720002,894997357851,
%T 45627309530399,2326097788692498,118585359913786999,
%U 6045527257814444451,308203304788622880002,15712323016961952435651,801020270560270951338199
%N Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at x=51/2.
%C a(n) and b(n):= A097836(n-1) with b(0) = 0 are the improper and proper nonnegative solutions of the Pell equation a(n)^2 - 53*(7*b(n))^2 = +4. - _Wolfdieter Lang_, Jun 27 2013
%H Indranil Ghosh, <a href="/A099368/b099368.txt">Table of n, a(n) for n = 0..584</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (51, -1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = 51*a(n-1) - a(n-2), n >= 1; a(-1)=51, a(0)=2.
%F a(n) = S(n, 51) - S(n-2, 51) = 2*T(n, 51/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 51)=A097836(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.
%F a(n)= ap^n + am^n, with ap:=(51 + 7*sqrt(53))/2 and am:=(51 - 7*sqrt(53))/2.
%F G.f.: (2-51*x)/(1-51*x+x^2).
%t LinearRecurrence[{51, -1}, {2, 51}, 15] (* or *) CoefficientList[Series[(2 - 51 x)/(1 - 51 x + x^2), {x, 0, 14}], x] (* _Michael De Vlieger_, Feb 08 2017 *)
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Oct 18 2004
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