A131216 Numbers X such that 99*X^2 - 2178 is a square.

11, 209, 4169, 83171, 1659251, 33101849, 660377729, 13174452731, 262828676891, 5243399085089, 104605153024889, 2086859661412691, 41632588075228931, 830564901843165929, 16569665448788089649, 330562744073918627051

Offset: 1

Author

Richard Choulet, Sep 27 2007

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (20,-1).

Crossrefs

Cf. A083043.

Programs

GAP

a:=[11,209];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 11*x*(1-x)/(1-20*x+x^2) )); // G. C. Greubel, Dec 06 2019

Maple

seq(coeff(series(11*x*(1-x)/(1-20*x+x^2), x, n+1), x, n), n = 0..20); # G. C. Greubel, Dec 06 2019

Mathematica

LinearRecurrence[{20, -1}, {11, 209}, 20] (* G. C. Greubel, Dec 06 2019 *)

PARI

my(x='x+O('x^20)); Vec(11*x*(1-x)/(1-20*x+x^2)) \\ G. C. Greubel, Dec 06 2019

Sage

def A131216_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 11*x*(1-x)/(1-20*x+x^2) ).list()
A131216_list(20) # G. C. Greubel, Dec 06 2019

Formula

a(n+2) = 20*a(n+1) - a(n).

a(n+1) = 10*a(n+1)+ sqrt(99*a(n)^2 -2178).

G.f.: 11*z*(1-z)/(1-20*z+z^2) - Richard Choulet, Oct 09 2007

a(n) = 11*A075839(n). - R. J. Mathar, Aug 22 2012