A083043 Integers y such that 11*x^2 - 9*y^2 = 2 for some integer x.

1, 21, 419, 8359, 166761, 3326861, 66370459, 1324082319, 26415275921, 526981436101, 10513213446099, 209737287485879, 4184232536271481, 83474913437943741, 1665314036222603339, 33222805811014123039

Offset

1,2

Links

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).

Formula

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

Maple

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

Mathematica

LinearRecurrence[{20, -1}, {1, 21}, 20] (* Harvey P. Dale, Jun 02 2014 *)

Prog

(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

Crossrefs

Cf. A075839, A075843.

Sequence in context: A358698 A020534 A106656 * A162807 A097833 A163145

Adjacent sequences: A083040 A083041 A083042 * A083044 A083045 A083046

Keyword

easy,nonn

Author

Michael Somos, Apr 17 2003

Extensions

Corrected by T. D. Noe, Nov 07 2006

Offset changed to 1 by G. C. Greubel, Dec 06 2019

Status

approved