A097780 Chebyshev polynomials S(n,25) with Diophantine property.

1, 25, 624, 15575, 388751, 9703200, 242191249, 6045078025, 150884759376, 3766073906375, 94000962899999, 2346257998593600, 58562449001940001, 1461714967049906425, 36484311727245720624, 910646078214093109175

Offset

0,2

Comments

All positive integer solutions of Pell equation b(n)^2 - 621*a(n)^2 = +4 together with b(n)=A090733(n+1), n>=0. Note that D=621=69*3^2 is not squarefree.

For positive n, a(n) equals the permanent of the tridiagonal matrix with 25'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,...,24}. - Milan Janjic, Jan 25 2015

Links

Indranil Ghosh, Table of n, a(n) for n = 0..714

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (25,-1).

Formula

a(n) = S(n, 25)=U(n, 25/2) = S(2*n+1, sqrt(25))/sqrt(25) 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) = 25*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=25; a(-1)=0.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.

G.f.: 1/(1-25*x+x^2).

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*24^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/23*(23 + 3*sqrt(69)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/50*(23 + 3*sqrt(69)). - Peter Bala, Dec 23 2012

Example

(x,y) = (2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.

Mathematica

LinearRecurrence[{25, -1}, {1, 25}, 20] (* Harvey P. Dale, Aug 23 2021 *)

Prog

(Sage) [lucas_number1(n, 25, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008

Crossrefs

Sequence in context: A307145 A061614 A171330 * A366476 A209222 A207691

Adjacent sequences: A097777 A097778 A097779 * A097781 A097782 A097783

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved