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%I #37 Sep 08 2022 08:45:14
%S 1,26,701,18901,509626,13741001,370497401,9989688826,269351100901,
%T 7262490035501,195817879857626,5279820266120401,142359329305393201,
%U 3838422070979496026,103495036587140999501,2790527565781827490501
%N First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.
%C (5*b(n))^2 - 29*a(n)^2 = -4 with b(n)=A097834(n) give all positive solutions of this Pell equation.
%H Indranil Ghosh, <a href="/A097835/b097835.txt">Table of n, a(n) for n = 0..697</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 (27,-1).
%F a(n) = ((-1)^n)*S(2*n, 5*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
%F G.f.: (1-x)/(1-27*x+x^2).
%F a(n) = S(n, 27) - S(n-1, 27) = T(2*n+1, sqrt(29)/2)/(sqrt(29)/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) = 27*a(n-1) - a(n-2), a(0)=1, a(1)=26. - _Philippe Deléham_, Nov 18 2008
%e All positive solutions of Pell equation x^2 - 29*y^2 = -4 are (5=5*1,1), (140=5*28,26), (3775=5*755,701), (101785=5*20357,18901), ...
%t LinearRecurrence[{27,-1},{1,26},30] (* _Harvey P. Dale_, May 31 2013 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-27*x+x^2)) \\ _G. C. Greubel_, Jan 12 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-27*x+x^2) )); // _G. C. Greubel_, Jan 12 2019
%o (Sage) ((1-x)/(1-27*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019
%o (GAP) a:=[1,26];; for n in [3..30] do a[n]:=27*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 12 2019
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 10 2004
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