A201227 a(n) = (A201225(n))^3 - (A201226(n))^2.

219375, 4566375, 82569375, 1482276375, 26598999375, 477300306375, 8564807109375, 153689228256375, 2757841302099375, 49487454210126375, 888016334480769375, 15934806566444316375, 285938501861517519375, 5130958226940871626375, 92071309583074172349375

Offset

1,1

Comments

Values d of solutions (x,y,d) of x^3-y^2 = d with decreasing coefficient r=sqrt(x)/d which r tend to 1/(1350*sqrt(5)) when d tends to infinity.

Also infinity family of solutions Mordell curve with extension sqrt(5) (another than A200218).

Conjecture: No more infinite families of solutions Mordell curves with extension sqrt(5) than A201227 and A200218.

Ratio a(n+1)/a(n) tends to 9+4*sqrt(5) when n tends to infinity.

Because all values in this sequence are positive, it means that A201225, A201226 and A201227 are even indexes subset of another sequence.

Links

Table of n, a(n) for n=1..15.

Index entries for linear recurrences with constant coefficients, signature (19, -19, 1).

Formula

a(n) = (A201225(n))^3 - (A201226(n))^2.

a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3).

G.f.: x*(3375*(-65-118*x+7*x^2))/((-1+x)*(1-18*x+x^2)).

a(n) = 3375*(-11-(-2+sqrt(5))*(9+4*sqrt(5))^(-n)+(2+sqrt(5))*(9+4*sqrt(5))^n). - Colin Barker, Mar 03 2016

Mathematica

LinearRecurrence[{19, -19, 1}, {219375, 4566375, 82569375}, 30] (* Harvey P. Dale, Sep 25 2012 *)

Crossrefs

Sequence in context: A251480 A072189 A187865 * A185531 A258678 A254912

Adjacent sequences: A201224 A201225 A201226 * A201228 A201229 A201230

Keyword

nonn

Author

Artur Jasinski, Nov 28 2011

Status

approved