A097835 First differences of Chebyshev polynomials S(n,27) = A097781(n) with Diophantine property.

1, 26, 701, 18901, 509626, 13741001, 370497401, 9989688826, 269351100901, 7262490035501, 195817879857626, 5279820266120401, 142359329305393201, 3838422070979496026, 103495036587140999501, 2790527565781827490501

Offset: 0

Author

Wolfdieter Lang, Sep 10 2004

Keywords

• nonn
• easy

Comments

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

Examples

			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), ...
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..697
• Tanya Khovanova, Recursive Sequences
• Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (27,-1).

Crossrefs

Cf. similar sequences listed in A238379.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-27*x+x^2) )); // G. C. Greubel, Jan 12 2019

Mathematica

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

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-27*x+x^2)) \\ G. C. Greubel, Jan 12 2019

Sage

((1-x)/(1-27*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

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. - Philippe Deléham, Nov 18 2008