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%I #41 Aug 30 2022 14:11:32
%S 1,52,2651,135149,6889948,351252199,17906972201,912904330052,
%T 46540213860451,2372638002552949,120957997916339948,
%U 6166485255730784399,314369790044353664401,16026692807006306100052,817046963367277257438251
%N Chebyshev polynomials S(n,51) + S(n-1,51) with Diophantine property.
%C (7*a(n))^2 - 53*b(n)^2 = -4 with b(n)=A097838(n) gives all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A097837/b097837.txt">Table of n, a(n) for n = 0..584</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 (51,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = S(n, 51) + S(n-1, 51) = S(2*n, 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). S(n, 51)=A097836(n).
%F a(n) = (-2/7)*i*((-1)^n)*T(2*n+1, 7*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)/(1-51*x+x^2).
%F a(n) = 51*a(n-1) - a(n-2); a(0)=1, a(1)=52. - _Philippe Deléham_, Nov 18 2008
%F From _Peter Bala_, Aug 26 2022: (Start)
%F a(n) = (2/7)*(7/2 o 7/2 o ... o 7/2) (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, 52, 0, 2651, 0, 135149, 0, ...], with o.g.f. x*(1 + x^2)/(1 - 51*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 = -49, 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,7) + (1/7)*( 1 + (-1)^(n+1) )*F(n+1,7), 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} 14*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 14*A054413(n)*x^n.
%F Exp( Sum_{n >= 1} (-14)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 14*A054413(n)*(-x)^n. (End)
%e All positive solutions of Pell equation x^2 - 53*y^2 = -4 are (7=7*1,1), (364=7*52,50), (18557=7*2651,2549), (946043=7*135149,129949), ...
%t LinearRecurrence[{51,-1}, {1,52}, 30] (* _G. C. Greubel_, Jan 12 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+x)/(1-51*x+x^2)) \\ _G. C. Greubel_, Jan 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-51*x+x^2) )); // _G. C. Greubel_, Jan 12 2019
%o (Sage) ((1+x)/(1-51*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019
%o (GAP) a:=[1,52];; for n in [3..30] do a[n]:=51*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 12 2019
%Y Cf. A097783, A097834, A097836, A097840.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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