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A097781 Chebyshev polynomials S(n,27) with Diophantine property. 7
1, 27, 728, 19629, 529255, 14270256, 384767657, 10374456483, 279725557384, 7542215592885, 203360095450511, 5483180361570912, 147842509666964113, 3986264580646460139, 107481301167787459640, 2898008866949614950141 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree.

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 27'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,...,26}. - Milan Janjic, Jan 26 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..700

R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (27,-1).

FORMULA

a(n) = S(n, 27) = U(n, 27/2) = S(2*n+1, sqrt(29))/sqrt(29) 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) = 27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (27+5*sqrt(29))/2 and am = (27-5*sqrt(29))/2.

G.f.: 1/(1-27*x+x^2).

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(29)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 5/54*(5 + sqrt(29)). - Peter Bala, Dec 23 2012

EXAMPLE

(x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4.

MAPLE

with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); # Zerinvary Lajos, Apr 20 2008

MATHEMATICA

Join[{a=1, b=27}, Table[c=27*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)

CoefficientList[Series[1/(1 - 27 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)

PROG

(Sage) [lucas_number1(n, 27, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) I:=[1, 27, 728]; [n le 3 select I[n] else 27*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012

CROSSREFS

Cf. A078362, A078366.

Sequence in context: A170746 A218729 A171332 * A223656 A073537 A016947

Adjacent sequences:  A097778 A097779 A097780 * A097782 A097783 A097784

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified August 27 18:12 EDT 2016. Contains 275912 sequences.