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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

Index entries for linear recurrences with constant coefficients, signature (27, -1).

Index entries for sequences related to Chebyshev polynomials.

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(-1-n) = -a(n). - 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-Fran├žois Alcover, Jun 21 2013, from 2nd formula *)

PROG

(PARI) {a(n) = (-1)^n * subst(2 * I / 5 * poltchebi(2*n), '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 April 29 17:20 EDT 2017. Contains 285607 sequences.