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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

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Last modified June 29 08:41 EDT 2017. Contains 288859 sequences.