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 A098253 First differences of Chebyshev polynomials S(n,363)=A098251(n) with Diophantine property. 4

%I

%S 1,362,131405,47699653,17314842634,6285240176489,2281524869222873,

%T 828187242287726410,300629687425575463957,109127748348241605689981,

%U 39613072020724277289999146,14379436015774564414664000017

%N First differences of Chebyshev polynomials S(n,363)=A098251(n) with Diophantine property.

%C (19*b(n))^2 - 365*a(n)^2 = -4 with b(n)=A098252(n) give all positive solutions of this Pell equation.

%H Indranil Ghosh, <a href="/A098253/b098253.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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (363,-1).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

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

%F a(n) = ((-1)^n)*S(2*n, 19*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.

%F a(n) = S(n, 363) - S(n-1, 363) = T(2*n+1, sqrt(365)/2)/(sqrt(365)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.

%F a(n) = 363*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=362. - _Philippe DelĂ©ham_, Nov 18 2008

%e All positive solutions of Pell equation x^2 - 365*y^2 = -4 are

%e (19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),

%e (911300591=19*47963189,47699653), ...

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 10 2004

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Last modified April 19 11:04 EDT 2019. Contains 322260 sequences. (Running on oeis4.)