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 A097780 Chebyshev polynomials S(n,25) with Diophantine property. 3
 1, 25, 624, 15575, 388751, 9703200, 242191249, 6045078025, 150884759376, 3766073906375, 94000962899999, 2346257998593600, 58562449001940001, 1461714967049906425, 36484311727245720624, 910646078214093109175 (list; graph; refs; listen; history; text; internal format)
 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 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 * A209222 A207691 A207926 Adjacent sequences:  A097777 A097778 A097779 * A097781 A097782 A097783 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified October 5 14:03 EDT 2022. Contains 357258 sequences. (Running on oeis4.)