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A097780
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Chebyshev polynomials S(n,25) with Diophantine property.
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2
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1, 25, 624, 15575, 388751, 9703200, 242191249, 6045078025, 150884759376, 3766073906375, 94000962899999, 2346257998593600, 58562449001940001, 1461714967049906425, 36484311727245720624, 910646078214093109175
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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). [From John M. Campbell, Jul 08 2011]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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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. - DELEHAM Philippe, Feb 10 2012
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EXAMPLE
| (x,y) = (2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.
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PROG
| sage: [lucas_number1(n, 25, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| Sequence in context: A170744 A061614 A171330 * A207345 A207268 A207023
Adjacent sequences: A097777 A097778 A097779 * A097781 A097782 A097783
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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