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A097783
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Chebyshev polynomials S(n,11) + S(n-1,11) with Diophantine property.
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7
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1, 12, 131, 1429, 15588, 170039, 1854841, 20233212, 220710491, 2407582189, 26262693588, 286482047279, 3125039826481, 34088956044012, 371853476657651, 4056299287190149, 44247438682433988, 482665526219583719
(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 (3*a(n))^2 - 13*b(n)^2 = -4 together with b(n)=A078922(n+1), n>=0.
a(n) = L(n,-11)*(-1)^n, where L is defined as in A108299; see also A078922 for L(n,+11). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
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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, 11) +S(n-1, 11) = S(2*n, sqrt(13)), 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)= (-2/3)*I*((-1)^n)*T(2*n+1, 3*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-11*x+x^2).
a(n)=11*a(n-1)-a(n-2) with a(0)=1 and a(1)=12. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
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EXAMPLE
| All positive solutions to the Pell equation x^2 - 13*y^2 = -4 are (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...
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PROG
| (Other) sage: [(lucas_number2(n, 11, 1)-lucas_number2(n-1, 11, 1))/9 for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]
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CROSSREFS
| Cf. S(n, 11)=A004190(n).
Sequence in context: A163414 A111777 A160962 * A078218 A048643 A111085
Adjacent sequences: A097780 A097781 A097782 * A097784 A097785 A097786
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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