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A083043
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Integers y such that 11*x^2 - 9*y^2 = 2 for some integer x.
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5
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1, 21, 419, 8359, 166761, 3326861, 66370459, 1324082319, 26415275921, 526981436101, 10513213446099, 209737287485879, 4184232536271481, 83474913437943741, 1665314036222603339, 33222805811014123039
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OFFSET
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1,2
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..750
Tanya Khovanova, Recursive Sequences
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (20,-1).
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FORMULA
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G.f.: x*(1+x)/(1-20*x+x^2).
a(n) = 20*a(n-1) - a(n-2).
a(1-n) = -a(n).
11*A075839(n)^2 - 9*a(n)^2 = 2.
a(n+1) = 10*a(n) + sqrt(99*a(n)^2 + 22). - Richard Choulet, Sep 27 2007
a(n) = ((3 + sqrt(11))*(10 + 3*sqrt(11))^(n-1) + (3 - sqrt(11))*(10 - 3*sqrt(11))^(n-1))/6. - G. C. Greubel, Dec 06 2019
E.g.f.: 1 + (1/3)*exp(10*x)*(-3*cosh(3*sqrt(11)*x) + sqrt(11)*sinh(3*sqrt(11)*x)). - Stefano Spezia, Dec 06 2019 after G. C. Greubel
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MAPLE
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seq(coeff(series( x*(1+x)/(1-20*x+x^2), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019
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MATHEMATICA
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LinearRecurrence[{20, -1}, {1, 21}, 20] (* Harvey P. Dale, Jun 02 2014 *)
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PROG
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(PARI) a(n)=subst(poltchebi(n+1)-poltchebi(n), x, 10)/9
(Magma) I:=[1, 21]; [n le 2 select I[n] else 20*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 06 2019
(Sage)
def A083043_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1+x)/(1-20*x+x^2) ).list()
a=A083043_list(20); a[1:] # G. C. Greubel, Dec 06 2019
(GAP) a:=[1, 21];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019
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CROSSREFS
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Cf. A075839, A075843.
Sequence in context: A358698 A020534 A106656 * A162807 A097833 A163145
Adjacent sequences: A083040 A083041 A083042 * A083044 A083045 A083046
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KEYWORD
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easy,nonn
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AUTHOR
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Michael Somos, Apr 17 2003
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EXTENSIONS
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Corrected by T. D. Noe, Nov 07 2006
Offset changed to 1 by G. C. Greubel, Dec 06 2019
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STATUS
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approved
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