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A078922
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a(n) = 11*a(n-1) - a(n-2).
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9
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1, 10, 109, 1189, 12970, 141481, 1543321, 16835050, 183642229, 2003229469, 21851881930, 238367471761, 2600190307441, 28363725910090, 309400794703549, 3375045015828949, 36816094379414890, 401601993157734841
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| All positive integer solutions of Pell equation (3*b(n))^2 - 13*a(n)^2 = -4 together with b(n)=A097783(n-1), n>=1.
a(n) = L(n-1,11), where L is defined as in A108299; see also A097783 for L(n,-11). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9, A} which do not end in 0. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jan 10 2007
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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(1)=1, a(2)=10 and for n>2 a(n)=ceiling(g*f^n) where f=(11+sqrt(117))/2 and g=(1-3/sqrt(13))/2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 12 2003
a(n)a(n+3) = 99 + a(n+1)a(n+2). - R. Stephan, May 29 2004
a(n) = S(n-1, 11) - S(n-2, 11) = T(2*n-1, sqrt(13)/2)/(sqrt(13)/2).
a(n+1)= ((-1)^n)*S(2*n, I*3), n>=0, with the imaginary unit I and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310.
G.f.: x*(1-x)/(1-11*x+x^2).
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EXAMPLE
| All positive solutions of the Pell equation x^2 - 13*y^2 = -4 are
(x,y)= (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...
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CROSSREFS
| Row 11 of array A094954.
Sequence in context: A024527 A198700 A015591 * A199760 A082181 A190919
Adjacent sequences: A078919 A078920 A078921 * A078923 A078924 A078925
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KEYWORD
| nonn
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AUTHOR
| Nick Renton (ner(AT)nickrenton.com), Jan 11 2003
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EXTENSIONS
| More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 12 2003
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