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A097835 First differences of Chebyshev polynomials S(n,27)=A097781(n) with Diophantine property. 5
1, 26, 701, 18901, 509626, 13741001, 370497401, 9989688826, 269351100901, 7262490035501, 195817879857626, 5279820266120401, 142359329305393201, 3838422070979496026, 103495036587140999501, 2790527565781827490501 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(5*b(n))^2 - 29*a(n)^2 = -4 with b(n)=A097834(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..697

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

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.

G.f.: (1-x)/(1-27*x+x^2).

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.

a(n)=27*a(n-1)-a(n-2), a(0)=1, a(1)=26. [From Philippe Deléham, Nov 18 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

LinearRecurrence[{27, -1}, {1, 26}, 30] (* Harvey P. Dale, May 31 2013 *)

CROSSREFS

Cf. similar sequences listed in A238379.

Sequence in context: A041313 A042302 A031423 * A158643 A181227 A094738

Adjacent sequences:  A097832 A097833 A097834 * A097836 A097837 A097838

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified April 27 18:42 EDT 2017. Contains 285528 sequences.