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A098252 Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property. 3

%I #32 Jan 22 2020 14:39:29

%S 1,364,132131,47963189,17410505476,6319965524599,2294130074923961,

%T 832762897231873244,302290637565095063611,109730668673232276217549,

%U 39831930437745751171906676,14458881018233034443125905839

%N Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property.

%C (19*a(n))^2 - 365*b(n)^2 = -4 with b(n)=A098253(n) give all positive solutions of this Pell equation.

%H Indranil Ghosh, <a href="/A098252/b098252.txt">Table of n, a(n) for n = 0..389</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%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 (363,-1).

%F a(n) = S(n, 363) + S(n-1, 363) = S(2*n, sqrt(365)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 363)=A098251(n).

%F a(n) = (-2/19)*i*((-1)^n)*T(2*n+1, 19*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

%F G.f.: (1+x)/(1-363*x+x^2).

%F a(n) = 363*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=364. - _Philippe Deléham_, Nov 18 2008

%e All positive solutions of Pell equation x^2 - 365*y^2 = -4 are (19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),(911300591=19*47963189,47699653), ...

%t LinearRecurrence[{363,-1},{1,364},20] (* _Harvey P. Dale_, Feb 03 2015 *)

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 10 2004

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Last modified March 28 14:33 EDT 2024. Contains 371254 sequences. (Running on oeis4.)