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.

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A200936 Successive values x of solutions Mordell's elliptic curve x^3-y^2 = d contained points {x,y} with quadratic extension sqrt(2) over rationals.

Original entry on oeis.org

22, 190, 2878, 3862, 111382, 117118, 3864190, 3897622, 131738902, 131933758, 4477986238, 4479121942, 152135692822, 152142312190, 5168228240638, 5168266821142, 175568164615702, 175568389479358, 5964152516784190, 5964153827385622, 202605635754466582
Offset: 0

Views

Author

Artur Jasinski, Nov 25 2011

Keywords

Comments

This sequence is equivalent of A200216, but A200216 was for quadratic field with extension sqrt(5).
Coefficient r=sqrt(x)/d tend to sqrt(2)/432 ~ 0.00327364 when x and d tend to infinity.
Starting from a(2)= 2878 all points are extremal (for definition see A200656).
(a(n)+10)/2 is perfect square of even number for each n.
All numbers in this sequence are of the form 2*(12*k+11).
For y values see A200937.
For d values see A200938.
When n is even d=A200938(n) are positive~, when n is odd d=A200938(n) are negative.

Examples

			a(3)=2878=A200656(1) because 2878^3-154396^2=15336.
G.f. = 22 + 190*x + 2868*x^2 + 3862*x^3 + 111382*x^4 + 117118*x^5 + ... - _Michael Somos_, Aug 23 2018

Links

Crossrefs

Cf. A001333, A200216, A200656.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)))); // G. C. Greubel, Jul 27 2018
  • Mathematica
    aa = {22, 190, 2878, 3862, 111382, 117118, 3864190, 3897622, 131738902}; a1 = aa[[1]]; a2 = aa[[3]]; a3 = aa[[3]]; a4 = aa[[4]]; a5 = aa[[5]]; a6 = aa[[6]]; a7 = aa[[7]]; a8 = aa[[8]]; a9 = aa[[9]]; Do[an = a9 + 40*a8 - 40*a7 - 206*a6 + 206*a5 + 40*a4 - 40*a3 - a2 + a1; a1 = a2; a2 = a3; a3 = a4; a4 = a5; a5 = a6; a6 = a7; a7 = a8; a8 = a9; a9 = an; AppendTo[aa, an], {nn, 20}]; aa
CoefficientList[Series[-2*(11 + 84*z + 904*z^2 - 2868*z^3 + 492*z^5 - 12*z^7 + 2266*z^4 - 440*z^6 + 11*z^8)/((z - 1) (z^2 + 6*z + 1) (1 - 6*z + z^2) (z^2 + 2*z - 1) (z^2 - 2*z - 1)), {z, 0, 30}], z] (* G. C. Greubel, Jul 27 2018 *)
a[ n_] := With[{m = Max[-5 - n, n]}, SeriesCoefficient[ 2 (1 - 12 x - 40 x^2 + 396 x^3 - 1138 x^4 + 396 x^5 - 40 x^6 - 12 x^7 + x^8) / (x^2 (x - 1) (1 + 6 x + x^2) (1 - 6 x + x^2) (x^2 + 2 x - 1) (x^2 - 2 x - 1)), {x, 0, m}]]; (* Michael Somos, Aug 23 2018 *)
a[ n_] := With[ {m = If[ OddQ[n], -5 - n, n], r1 = 1 + Sqrt[2], r2 = 1 - Sqrt[2]}, Simplify[7 - 6 (6 r1 + r2) r1^m - 6 (r1 + 6 r2) r2^m + (169 r1 + 29 r2)/4 r1^(2 m) + (29 r1 + 169 r2)/4 r2^(2 m)]]; (* Michael Somos, Aug 25 2018 *)
  • PARI
    z='z+O('z^30); Vec(2*(11+84*z+904*z^2-2868*z^3+492*z^5 -12*z^7 +2266*z^4 -440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1))) \\ G. C. Greubel, Jul 27 2018
  • PARI
    {a(n) = my(m = max(-5-n, n)); polcoeff( 2*(1 - 12*x - 40*x^2 + 396*x^3 - 1138*x^4 + 396*x^5 - 40*x^6 - 12*x^7 + x^8) / (x^2*(x - 1)*(1 + 6*x + x^2)*(1 - 6*x + x^2)*(x^2 + 2*x - 1)*(x^2 - 2*x - 1)) + x * O(x^m), m)}; /* Michael Somos, Aug 23 2018 */
  • PARI
    {a(n) = my(m = if(n%2, -5-n, n), r1 = 1 + quadgen(8), r2 = 1 - quadgen(8)); simplify(7 - 6*(6*r1 + r2) * r1^m - 6*(r1 + 6*r2) * r2^m + (169*r1 + 29*r2)/4 * r1^(2*m) + (29*r1 + 169*r2)/4 * r2^(2*m))}; /* Michael Somos, Aug 25 2018 */

Formula

a(n) = (A200937(n)^2 + A200938(n))^(1/3).
a(n) = a(n-1)+ 40*a(n-2) - 40*a(n-3) - 206*a(n-4) + 206*a(n-5) + 40*a(n-6) - 40*a(n-7) - a(n-8) + a(n-9).
G.f.: 2*(11+84*z+904*z^2-2868*z^3+492*z^5-12*z^7+2266*z^4-440*z^6 +11*z^8)/((1-z)*(z^2+6*z+1)*(1-6*z+z^2)*(z^2+2*z-1)*(z^2-2*z-1)).
a(2*n + 1) - a(2*n) = 24 * A001333(2*n + 3), a(n) = a(-5-n) for all n in Z. - Michael Somos, Aug 23 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
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