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

Author

Artur Jasinski, Nov 28 2011

Keywords

• nonn

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

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

Programs

Mathematica

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

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