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%I #43 Aug 30 2022 14:11:40
%S 1,84,6971,578509,48009276,3984191399,330639876841,27439125586404,
%T 2277116783794691,188973253929372949,15682502959354160076,
%U 1301458772372465913359,108005395603955316648721
%N Chebyshev polynomials S(n,83) + S(n-1,83) with Diophantine property.
%C (9*a(n))^2 - 85*b(n)^2 = -4 with b(n)=A097841(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A097840/b097840.txt">Table of n, a(n) for n = 0..520</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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (83, -1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = S(n, 83) + S(n-1, 83) = S(2*n, sqrt(85)), 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, 83) = A097839(n).
%F a(n) = (-2/9)*i*((-1)^n)*T(2*n+1, 9*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 - 83*x + x^2).
%F a(n) = 83*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=84. - _Philippe Deléham_, Nov 18 2008
%F From _Peter Bala_, Aug 26 2022: (Start)
%F a(n) = (2/9)*(9/2 o 9/2 o ... o 9/2) (2*n+1 terms), where the binary operation o is defined on real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0.
%F The aerated sequence (b(n))n>=1 = [1, 0, 84, 0, 6971, 0, 578509, 0, ...], with o.g.f. x*(1 + x^2)/(1 - 83*x^2 + x^4), is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -81, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials.
%F b(n) = 1/2*( (-1)^n - 1 )*F(n,9) + 1/9*( 1 + (-1)^(n+1) )*F(n+1,9), where F(n,x) is the n-th Fibonacci polynomial - see A168561 (but with row indexing starting at n = 1).
%F Exp( Sum_{n >= 1} 18*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 18*A099371(n)*x^n.
%F Exp( Sum_{n >= 1} (-18)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 18*A099371(n)*(-x)^n. (End)
%e All positive solutions of Pell equation x^2 - 85*y^2 = -4 are (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...
%t CoefficientList[Series[(1+x)/(1-83x+x^2), {x, 0, 20}], x] (* _Michael De Vlieger_, Feb 08 2017 *)
%o (PARI) my(x='x+O('x^20)); Vec((1+x)/(1-83*x+x^2)) \\ _G. C. Greubel_, Jan 13 2019
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-83*x+x^2) )); // _G. C. Greubel_, Jan 13 2019
%o (Sage) ((1+x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 13 2019
%o (GAP) a:=[1,84];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 13 2019
%Y Cf. A097783, A097834, A097837, A097839.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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