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A097778
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Chebyshev polynomials S(n,23) with Diophantine property.
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3
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1, 23, 528, 12121, 278255, 6387744, 146639857, 3366328967, 77278926384, 1774048977865, 40725847564511, 934920445005888, 21462444387570913, 492701300469125111, 11310667466402306640, 259652650426783927609
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OFFSET
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0,2
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COMMENTS
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All positive integer solutions of Pell equation b(n)^2 - 525*a(n)^2 = +4 together with b(n)=A090731(n+1), n>=0. Note that D=525=21*5^2 is not squarefree.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 23'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,...,22}. - Milan Janjic, Jan 25 2015
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 0..733
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 relate d to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (23,-1).
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FORMULA
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a(n) = S(n, 23) = U(n, 23/2) = 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) = 23*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=23; a(-1)=0.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
G.f.: 1/(1-23*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*22^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/21*(21 + 5*sqrt(21)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/46*(21 + 5*sqrt(21)). - Peter Bala, Dec 23 2012
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EXAMPLE
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(x,y) = (23;1), (527;23), (12098;528), ... give the positive integer solutions to x^2 - 21*(5*y)^2 =+4.
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MATHEMATICA
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LinearRecurrence[{23, -1}, {1, 23}, 20] (* Harvey P. Dale, May 06 2016 *)
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PROG
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(Sage) [lucas_number1(n, 23, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
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CROSSREFS
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Sequence in context: A114926 A118338 A171328 * A332797 A057193 A014960
Adjacent sequences: A097775 A097776 A097777 * A097779 A097780 A097781
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 31 2004
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STATUS
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approved
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