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%I #47 Mar 06 2021 01:31:35
%S 1,33,2177,143649,9478657,625447713,41270070401,2723199198753,
%T 179689877047297,11856808685922849,782369683393860737,
%U 51624542295308885793,3406437421806992601601,224773245296966202819873
%N Chebyshev polynomial of the first kind, T(n,x), evaluated at x=33.
%C Used in A099369.
%C Solutions of the Pell equation x^2 - 17y^2 = 1 (x values). After initial term this sequence bisects A041024. See 8*A097316(n-1) with A097316(-1) = 0 for corresponding y values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2. (See related comments in A088317, which this sequence also bisects.). - _Rick L. Shepherd_, Jul 31 2006
%C From a(n) = T(n, 33) (see the formula section) and the de Moivre-Binet formula for T(n,x=33) follows a(n+1)/a(n) = 33 + 8*sqrt(17), which is the conjectured value (4+sqrt(17))^2 given in the previous comment by Rick L. Shepherd. - _Wolfdieter Lang_, Jun 28 2013
%H Vincenzo Librandi, <a href="/A099370/b099370.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation</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 (66,-1).
%F a(n) = 66*a(n-1) - a(n-2), a(-1):= 33, a(0)=1.
%F a(n) = T(n, 33) = (S(n, 66)-S(n-2, 66))/2 = S(n, 66)-33*S(n-1, 66) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp.second, kind. See A053120 and A049310. S(n, 66)=A097316(n).
%F a(n) = ((33+8*sqrt(17))^n + (33-8*sqrt(17))^n)/2.
%F a(n) = Sum_{k=0..floor(n/2)} ((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*33)^(n-2*k), for n>=1, a(0)=1.
%F G.f.: (1-33*x)/(1-66*x+x^2).
%e a(1)^2 - 17*A121470(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.
%t LinearRecurrence[{66, -1},{1, 33},14] (* _Ray Chandler_, Aug 11 2015 *)
%o (PARI) \\ Program uses fact that continued fraction for sqrt(17) = [4,8,8,...].
%o print1("1, "); forstep(n=2,40,2,v=vector(n,i,if(i>1,8,4)); print1(contfracpnqn(v)[1,1],", ")) \\ _Rick L. Shepherd_, Jul 31 2006
%o (PARI) vector(20,n,polchebyshev(n-1,1,33)) \\ _Joerg Arndt_, Jan 01 2021
%Y Cf. A121470, A041024, A040012.
%Y Row 4 of array A188645.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 18 2004
%E A-number for y values in Pell equation corrected by _Wolfdieter Lang_, Jun 28 2013
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