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 A097834 Chebyshev polynomials S(n,27) + S(n-1,27) with Diophantine property. 3
 1, 28, 755, 20357, 548884, 14799511, 399037913, 10759224140, 290100013867, 7821941150269, 210902311043396, 5686540457021423, 153325690028535025, 4134107090313424252, 111467565748433919779, 3005490168117402409781 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS (5*a(n))^2 - 29*b(n)^2 = -4 with b(n)=A097835(n) give all positive solutions of this Pell equation. LINKS Indranil Ghosh, Table of n, a(n) for n = 0..697 (terms 0..200 from Vincenzo Librandi) Tanya Khovanova, Recursive Sequences Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16. Index entries for linear recurrences with constant coefficients, signature (27,-1). FORMULA a(n) = S(n, 27) + S(n-1, 27) = S(2*n, sqrt(29)), 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, 27)=A097781(n). a(n) = (-2/5)*i*((-1)^n)*T(2*n+1, 5*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120. G.f.: (1+x)/(1-27*x+x^2). a(n) = - a(-1-n) for all n in Z. - Michael Somos, Nov 01 2008 EXAMPLE 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), ... MATHEMATICA a[n_] := -2/5*I*(-1)^n*ChebyshevT[2*n + 1, 5*I/2]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 21 2013, from 2nd formula *) PROG (PARI) {a(n) = (-1)^n * subst(2 * I / 5 * poltchebi(2*n + 1), 'x, -5/2 * I)}; /* Michael Somos, Nov 04 2008 */ CROSSREFS A087130(2*n + 1) = 5 * a(n). - Michael Somos, Nov 01 2008 Sequence in context: A226991 A277060 A229463 * A162830 A163187 A163548 Adjacent sequences:  A097831 A097832 A097833 * A097835 A097836 A097837 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Sep 10 2004 STATUS approved

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Last modified October 29 21:59 EDT 2020. Contains 338074 sequences. (Running on oeis4.)