%I #32 Sep 08 2022 08:45:14

%S 1,51,2600,132549,6757399,344494800,17562477401,895341852651,

%T 45644872007800,2326993130545149,118631004785794799,

%U 6047854250944989600,308321935793408674801,15718370871212897425251,801328592496064360013000

%N Chebyshev polynomials S(n,51).

%C Used for all positive integer solutions of Pell equation x^2 - 53*y^2 = -4. See A097837 with A097838.

%C a(n-1), with a(-1) := 0, and b(n) := A099368(n) give the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 53*(7*a(n-1))^2 = +4, n >= 0. - _Wolfdieter Lang_, Jun 27 2013

%H Indranil Ghosh, <a href="/A097836/b097836.txt">Table of n, a(n) for n = 0..584</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 R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

%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) = S(n, 51)=U(n, 51/2)= S(2*n+1, sqrt(53))/sqrt(53) 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) = 51*a(n-1) - a(n-2), n >= 1, a(-1)=0, a(0)=1, a(1)=51.

%F a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (51+7*sqrt(53))/2 and am := (51-7*sqrt(53))/2 = 1/ap.

%F G.f.: 1/(1-51*x+x^2).

%t LinearRecurrence[{51,-1}, {1,51}, 30] (* _G. C. Greubel_, Jan 12 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(1/(1-51*x+x^2)) \\ _G. C. Greubel_, Jan 12 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(1-51*x+x^2) )); // _G. C. Greubel_, Jan 12 2019

%o (Sage) (1/(1-51*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019

%o (GAP) a:=[1,51];; for n in [2..30] do a[n]:=51*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 12 2019

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 10 2004