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A097842 Chebyshev polynomials S(n,123) + S(n-1,123) with Diophantine property.

%I A097842 #48 Sep 08 2022 08:45:14
%S A097842 1,124,15251,1875749,230701876,28374454999,3489827263001,
%T A097842 429220378894124,52790616776714251,6492816643156958749,
%U A097842 798563656491529211876,98216836931814936101999,12079872378956745611334001,1485726085774747895257980124,182732228677915034371120221251,22474578401297774479752529233749
%N A097842 Chebyshev polynomials S(n,123) + S(n-1,123) with Diophantine property.
%C A097842 (11*a(n))^2 - 5*(5*b(n))^2 = -4 with b(n)=A097843(n) give all positive solutions of this Pell equation.
%H A097842 Indranil Ghosh, <a href="/A097842/b097842.txt">Table of n, a(n) for n = 0..477</a>
%H A097842 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097842 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 A097842 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A097842 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html ">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A097842 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (123,-1).
%H A097842 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A097842 a(n) = S(n, 123) + S(n-1, 123) = S(2*n, 5*sqrt(5)), 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, 123) = A049670(n+1).
%F A097842 a(n) = (-2/11)*i*((-1)^n)*T(2*n+1, 11*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
%F A097842 G.f.: (1+x)/(1-123*x+x^2).
%F A097842 a(n) = 123*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=124. - _Philippe Deléham_, Nov 18 2008
%F A097842 From _Peter Bala_, Mar 23 2015: (Start)
%F A097842 a(n) = ( Fibonacci(10*n + 10 - 2*k) + Fibonacci(10*n + 2*k) )/( Fibonacci(10 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.
%F A097842 a(n) = ( Fibonacci(10*n + 10 - 2*k - 1) - Fibonacci(10*n + 2*k + 1) )/( Fibonacci(10 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer, k != 2.
%F A097842 The aerated sequence (b(n))n>=1 = [1, 0, 124, 0, 15251, 0, 1875749, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -121, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)
%F A097842 a(n) = Lucas(10*n + 5)/11. - _Ehren Metcalfe_, Jul 29 2017
%e A097842 All positive solutions of Pell equation x^2 - 125*y^2 = -4 are (11=11*1,1), (1364=11*124,122), (167761=11*15251,15005), (20633239=11*1875749,1845493), ...
%t A097842 CoefficientList[Series[(1+x)/(1-123x+x^2), {x, 0, 20}], x] (* _Michael De Vlieger_, Feb 08 2017 *)
%t A097842 LinearRecurrence[{123,-1}, {1,124}, 20] (* _G. C. Greubel_, Jan 13 2019 *)
%o A097842 (PARI) a(n)=polchebyshev(n, 2, 123/2) + polchebyshev(n - 1, 2, 123/2); \\ _Michel Marcus_, Aug 04 2017
%o A097842 (PARI) my(x='x+O('x^20)); Vec((1+x)/(1-123*x+x^2)) \\ _G. C. Greubel_, Jan 13 2019
%o A097842 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-123*x+x^2) )); // _G. C. Greubel_, Jan 13 2019
%o A097842 (Sage) ((1+x)/(1-123*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 13 2019
%o A097842 (GAP) a:=[1,124];; for n in [3..20] do a[n]:=123*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 13 2019
%K A097842 nonn,easy
%O A097842 0,2
%A A097842 _Wolfdieter Lang_, Sep 10 2004