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.

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.

A200937 Values y for infinite sequence x^3 - y^2 = d with increasing coefficient r = sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

Original entry on oeis.org

100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956, 59339881525800500, 59343754352533100, 11749314454296080876, 11749446016399614644, 2326315710145219660324, 2326320179383913075836, 460599127771776655165660, 460599279594330127759300
Offset: 0

Views

Author

Artur Jasinski, Nov 25 2011

Keywords

Comments

For x values see A200936.
For d values see A200938.
This sequence is equivalent of A200217, but A200217 was for quadratic field with extension sqrt(5).
All numbers in this sequence are of the form 4*(2*n+1).

Links

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((100 +2620*x +130596*x^2 -383556*x^3 +1239016*x^4 +4252504*x^5 -332600*x^6 -932360*x^7 +10356*x^8 +27564*x^9 -44*x^10 -116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 -14*x -1)))); // G. C. Greubel, Aug 22 2018
  • Mathematica
    aa = {100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956}; a1 = aa[[1]]; a2 = aa[[2]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; a6 = aa[[6]]; a7 = aa[[7]]; a8 = aa[[8]]; a9 = aa[[9]]; a10 = aa[[10]]; a11 = aa[[11]]; a12 = aa[[12]]; Do[an = 238*a11 - 8127*a9 + 40868*a7 - 8127*a5 + 238*a3 - a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = a6; a6 = a7; a7 = a8; a8 = a9; a9 = a10; a10 = a11; a11 = a12; a12 = an; AppendTo[aa, an], {nn, 1, 88}]; aa
LinearRecurrence[{0, 238, 0, -8127, 0, 40868, 0, -8127, 0, 238, 0, -1}, {100, 2620, 154396, 240004, 37172564, 40080716, 7596048140, 7694839700, 1512067083076, 1515423087964, 299656796131324, 299770801505956}, 50] (* G. C. Greubel, Aug 22 2018 *)
  • PARI
    x='x+O('x^30); Vec((100 +2620*x +130596*x^2 -383556*x^3 +1239016*x^4 +4252504*x^5 -332600*x^6 -932360*x^7 +10356*x^8 +27564*x^9 -44*x^10 -116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 -14*x -1))) \\ G. C. Greubel, Aug 18 2018

Formula

a(n) = sqrt(A200936(n)^3 - A200938(n)).
a(n) = 238*a(n-2) - 8127*a(n-4) + 40868*a(n-6) - 8127*a(n-8) + 238*a(n-10) - a(n-12).
G.f.: (100 + 2620*x + 130596*x^2 - 383556*x^3 + 1239016*x^4 + 4252504*x^5 - 332600*x^6 - 932360*x^7 + 10356*x^8 + 27564*x^9 - 44*x^10 - 116*x^11)/( (x^2+6*x+1)*(x^2-6*x+1)*(x^2+2*x-1)*(x^2-2*x-1)*(x^2+14*x-1)*(x^2 - 14*x - 1)). - R. J. Mathar, Nov 25 2011

Extensions

Data corrected by G. C. Greubel, Aug 22 2018

A200938 Values d for infinite sequence x^3-y^2 = d with increasing coefficient r=sqrt(x)/|d| or family of solutions Mordell curve with extension sqrt(2).

Original entry on oeis.org

648, -5400, 15336, -20088, 100872, -105624, 599400, -604152, 3505032, -3509784, 20440296, -20445048, 119146248, -119151000, 694446696, -694451448, 4047543432, -4047548184, 23590823400, -23590828152, 137497406472, -137497411224, 801393624936, -801393629688
Offset: 0

Views

Author

Artur Jasinski, Nov 25 2011

Keywords

Comments

For x values see A200936.
For y values see A200937.
This sequence is equivalent of A200218, but A200218 was for quadratic field with extension sqrt(5).
All numbers in this sequence are of the form 216*(4k+3).
When indices n are even d=a(n) are positive, when n is odd d=a(n) are negative.

Links

Crossrefs

Cf. A200216, A200217, A200218, A200936, A200937.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(216*(3-28*x+78*x^2+4*x^3-13*x^4)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2)))); // G. C. Greubel, Aug 18 2018
  • Mathematica
    uu = {648, -5400, 15336, -20088, 100872}; a1 = aa[[1]]; a2 = aa[[2]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; Do[an = a5 + 6 a4 - 6 a3 - a2 + a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = an; AppendTo[uu, an], {nn, 1, 20}]; uu
  • PARI
    my(x='x+O('x^30)); Vec(216*(3-28*x+78*x^2+4*x^3-13*x^4)/((1-x)*(1+2*x-x^2)*(1-2*x-x^2))) \\ G. C. Greubel, Aug 18 2018

Formula

a(n) = A200936(n)^3 - A200937(n)^2.
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - a(n-4) + a(n-5).
G.f.: 216*(3 - 28*z + 78*z^2 + 4*z^3 - 13*z^4)/((1 - z)*(1 + 2*z - z^2) *(1 - 2*z - z^2)).
E.g.f.: 216*(cosh(x)*(14*cosh(sqrt(2)*x) - 4*sqrt(2)*sinh(sqrt(2)*x) - 11) - sinh(x)*(6*cosh(sqrt(2)*x) - 10*sqrt(2)*sinh(sqrt(2)*x) + 11)). - Stefano Spezia, Oct 03 2022
Showing 1-3 of 3 results.

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