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%I #29 Sep 08 2022 08:45:14
%S 1,171,29240,4999869,854948359,146191169520,24997835039561,
%T 4274483600595411,730911697866775720,124981625851618052709,
%U 21371127108928820237519,3654337754000976642563040,624870384807058077058042321,106849181464252930200282673851
%N Chebyshev polynomials S(n,171).
%C Used for all positive integer solutions of Pell equation x^2 - 173*y^2 = -4. See A097845 with A098244.
%H Indranil Ghosh, <a href="/A097844/b097844.txt">Table of n, a(n) for n = 0..446</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 <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 (171,-1).
%F a(n) = S(n, 171) = U(n, 171/2) = S(2*n+1, sqrt(173))/sqrt(173) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x).
%F a(n) = 171*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=171.
%F a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (171+13*sqrt(173))/2 and am = (171-13*sqrt(173))/2 = 1/ap.
%F G.f.: 1/(1-171*x+x^2).
%t CoefficientList[Series[1/(1-171x+x^2),{x,0,30}],x] (* or *) LinearRecurrence[{171,-1},{1,171},30] (* _Harvey P. Dale_, Mar 21 2013 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-171*x+x^2)) \\ _G. C. Greubel_, Jan 14 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-171*x+x^2) )); // _G. C. Greubel_, Jan 14 2019
%o (Sage) (1/(1-171*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 14 2019
%o (GAP) a:=[1,171];; for n in [3..30] do a[n]:=171*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 14 2019
%Y Cf. A097845, A098244.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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