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).

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

0,1

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-1,1).

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

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

Prog

(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
(Magma) m:=30; R<x>:=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

Crossrefs

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

Sequence in context: A337313 A204392 A233677 * A165611 A282334 A034619

Adjacent sequences: A200935 A200936 A200937 * A200939 A200940 A200941

Keyword

sign,easy

Author

Artur Jasinski, Nov 25 2011

Status

approved