A097782 Chebyshev polynomials S(n,29) with Diophantine property.

1, 29, 840, 24331, 704759, 20413680, 591291961, 17127053189, 496093250520, 14369577211891, 416221645894319, 12056058153723360, 349209464812083121, 10115018421396687149, 292986324755691844200, 8486488399493666794651

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Comments

All positive integer solutions of Pell equation b(n)^2 - 837*a(n)^2 = +4 together with b(n)=A090251(n+1), n >= 0. Note that D=837=93*3^2 is not squarefree.

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 29's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,28}. - Milan Janjic, Jan 26 2015

Examples

			(x,y) = (29;1), (839;29), (24302,840), ..., give the positive integer solutions to x^2 - 93*(3*y)^2 =+4.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..682
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (29, -1).

Crossrefs

Cf. A097781.

Programs

GAP

a:=[1,29];; for n in [3..20] do a[n]:=29*a[n-1]-a[n-2]; od; a; # G. C. Greubel, May 25 2019

Magma

I:=[1,29]; [n le 2 select I[n] else 29*Self(n-1)-Self(n-2): n in [1..30]]; // G. C. Greubel, May 25 2019

Mathematica

LinearRecurrence[{29,-1},{1,29},20] (* Harvey P. Dale, Dec 14 2011 *)

PARI

my(x='x+O('x^30)); Vec(1/(1-29*x+x^2)) \\ G. C. Greubel, May 25 2019

Sage

[lucas_number1(n,29,1) for n in range(1,20)] # Zerinvary Lajos, Jun 27 2008

Formula

a(n) = S(n, 29) = U(n, 29/2) = S(2*n+1, sqrt(31))/sqrt(31) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).

a(n) = 29*a(n-1) - a(n-2), n >= 1; a(0)=1, a(1)=29; a(-1)=0.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (29+3*sqrt(93))/2 and am = (29-3*sqrt(93))/2.

G.f.: 1/(1-29*x+x^2).

a(n) = Sum_{k=0..n} A101950(n,k)*28^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/9*(9 + sqrt(93)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 3/58*(9 + sqrt(93)). - Peter Bala, Dec 23 2012