

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
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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 01avoiding words of length n1 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(n1)a(n2), n >= 1; a(0)=1, a(1)=29; a(1)=0.
a(n) = (ap^(n+1)  am^(n+1))/(apam) with ap = (29+3*sqrt(93))/2 and am = (293*sqrt(93))/2.
G.f.: 1/(129*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*ba; 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



