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

Author

Michael Somos, Apr 17 2003

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A075839, A075843.

Programs

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

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

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 *)

PARI

a(n)=subst(poltchebi(n+1)-poltchebi(n),x,10)/9

Sage

def A083043_list(prec):
    P. = 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

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

Extensions

Corrected by T. D. Noe, Nov 07 2006

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