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).

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

Offset

0,1

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

G. C. Greubel, Table of n, a(n) for n = 0..850

Index entries for linear recurrences with constant coefficients, signature (0,238,0,-8127,0,40868,0,-8127,0, 238,0,-1).

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

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 *)

Prog

(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
(Magma) m:=50; R<x>:=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

Crossrefs

Sequence in context: A128988 A075822 A250845 * A112889 A118490 A146310

Adjacent sequences: A200934 A200935 A200936 * A200938 A200939 A200940

Keyword

nonn

Author

Artur Jasinski, Nov 25 2011

Extensions

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

Status

approved