cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

219375, 4566375, 82569375, 1482276375, 26598999375, 477300306375, 8564807109375, 153689228256375, 2757841302099375, 49487454210126375, 888016334480769375, 15934806566444316375, 285938501861517519375, 5130958226940871626375, 92071309583074172349375
Offset: 1

Views

Author

Artur Jasinski, Nov 28 2011

Keywords

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

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

A202054 Smallest x such that Mordell's elliptic curve x^3-y^2=d (for positive d) has an integral point in the quadratic extension sqrt(A202057(n)).

Original entry on oeis.org

22, 6100, 88, 129910, 2860, 1193740, 2545, 6815614
Offset: 1

Views

Author

Artur Jasinski, Dec 10 2011

Keywords

Comments

The existence of such x for each A202057(n) is conjectural following the conjecture in A201278.
For y values see A202055.
For d values see A202056.
a(1) = A200936(1).
a(2) = A201225(1).

Crossrefs

Cf. A201278, A202054, A202055, A202056, A202057.

A201226 Values y for infinite sequence x^3-y^2 = d with decreasing coefficient r=sqrt(x)/d which tend to 1/(1350*sqrt(5))or infinity family of solutions Mordell curve with extension sqrt(5).

Original entry on oeis.org

476425, 3499913125, 20477027135825, 118398467411226125, 684132799477496491225, 3952927722012200964659125, 22840018438688230285823134625, 131969623469705492999569294932125, 762520461852550579069844280191898025
Offset: 1

Views

Author

Artur Jasinski, Nov 28 2011

Keywords

Comments

For x values see A201225.
For d values see A201227.

Links

Crossrefs

Cf. A201225, A201227.

Programs

  • Mathematica
    LinearRecurrence[{6118,-1970319,33501524,-1970319,6118,-1},{476425,3499913125,20477027135825,118398467411226125,684132799477496491225,3952927722012200964659125},20] (* Harvey P. Dale, Jun 12 2013 *)

Formula

G.f.: -((25*(-19057-23405799*z-130714666*z^2+9631442*z^3-30549*z^4+5*z^5))/((1-5778*z+z^2)*(1-322*z+z^2)*(1-18*z+z^2))).
a(n) = 6118*a(n-1) - 1970319*a(n-2) + 33501524*a(n-3) - 1970319*a(n-4) + 6118*a(n-5) - a(n-6).
Showing 1-3 of 3 results.

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