login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097782 Chebyshev polynomials S(n,29) with Diophantine property. 2
1, 29, 840, 24331, 704759, 20413680, 591291961, 17127053189, 496093250520, 14369577211891, 416221645894319, 12056058153723360, 349209464812083121, 10115018421396687149, 292986324755691844200, 8486488399493666794651 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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).

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<=k<=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

EXAMPLE

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

MATHEMATICA

Join[{a=1, b=29}, Table[c=29*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)

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

PROG

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

CROSSREFS

Cf. A097781

Sequence in context: A170748 A218731 A171334 * A223643 A223668 A223636

Adjacent sequences:  A097779 A097780 A097781 * A097783 A097784 A097785

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 20 04:05 EST 2017. Contains 294959 sequences.