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A097726
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Pell equation solutions (5*a(n))^2 - 26*b(n)^2 = -1 with b(n):=A097727(n), n>=0.
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2
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1, 103, 10505, 1071407, 109273009, 11144775511, 1136657829113, 115927953794015, 11823514629160417, 1205882564220568519, 122988198035868828521, 12543590317094399940623
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(-1)=-1 [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)= S(n, 2*51) + S(n-1, 2*51) = S(2*n, 2*sqrt(26)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n)= ((-1)^n)*T(2*n+1, 5*I)/(5*I) with the imaginary unit I and Chebyshev polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-102*x+x^2).
a(n)=102*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=103 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
a(n)=(1/5)Sinh[(2n-1)ArcSinh[5]] {n=1,2,3,...} [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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EXAMPLE
| (x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 =-1.
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MATHEMATICA
| Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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CROSSREFS
| Cf. A097725 for S(n, 102).
A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115,A173116 A173121. [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
Sequence in context: A034180 A076460 A172116 * A088584 A097014 A106297
Adjacent sequences: A097723 A097724 A097725 * A097727 A097728 A097729
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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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