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

Author

Artur Jasinski, Nov 25 2011

Keywords

• nonn

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).

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