A200216 -id:A200216 - cp's OEIS frontend

A200335 a(n) = sqrt((A200216(n)+1)/5).

137, 253772063, 472142416783537, 878420022140682133063, 1634298694352222684783778137, 3040609452244043180572708973082863, 5657047804679503550674811676317937783937, 10524926126507566387571141730985597902165021463

Offset: 1

Artur Jasinski, Nov 16 2011

nonn

All numbers (A200216(n)+1)/5 are perfect squares

Vincenzo Librandi, Table of n, a(n) for n = 1..50
Index entries for linear recurrences with constant coefficients, signature (1860497, 1860497, -1).

Cf. A200216, A200217, A200218.

I:=[137, 253772063, 472142416783537]; [n le 3 select I[n] else 1860497*Self(n-1)+1860497*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 18 2011

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, Sqrt[(xx + 1)/5]], {n, 1, 20}]; aa

x='x+O('x^30); Vec((137 -1116026*x +137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3)) \\ G. C. Greubel, Jul 10 2018

G.f.: (137 - 1116026*x + 137*x^2)/(1 - 1860497*x - 1860497*x^2 + x^3).

a(n) = 1860497*a(n-1) + 1860497*a(n-2) - a(n-3). [corrected by Vincenzo Librandi, Nov 18 2011]

A078933 Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.

Original entry on oeis.org

2, 5234, 8158, 93844, 367806, 421351, 720114, 939787, 28187351, 110781386, 154319269, 384242766, 390620082, 3790689201, 65589428378, 952764389446, 12438517260105, 35495694227489, 53197086958290, 5853886516781223, 12813608766102806, 23415546067124892, 38115991067861271

Offset: 1

Dean Hickerson and Robert G. Wilson v, Dec 16 2002

nonn

Hall conjectured that the nonzero difference k = x^3 - y^2 cannot be less than C x^(1/2), for a constant C. His original conjecture, probably false, has been reformulated in the following way: For any exponent e < 1/2, a constant K_e > 0 exists such that |x^3 - y^2| > K_e x^e.

Danilov found an infinite family of solutions to |x^3 - y^2| < sqrt(x). For more detail see A200216. [Artur Jasinski, Nov 04 2011]

			|5234^3 - 378661^2| = 17 < sqrt(5234), so 5234 is in the sequence.

Noam D. Elkies, Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction. Algorithmic Number Theory. Proceedings of ANTS-IV; W. Bosma, ed.; Springer, 2000; pp. 33-63.
Marshall Hall Jr., The Diophantine equation x^3 - y^2 = k, in Computers in Number Theory; A. O. L. Atkin and B. Birch, eds.; Academic Press, 1971; pp. 173-198.

Frank A. Stevenson, Table of n, a(n) for n = 1..54
S. Aanderaa, L. Kristiansen, and H. K. Ruud, Search for good examples of Hall's conjecture, Math. Comp. 87 (2018), 2903-2914.
Ismael Jimenez Calvo, Marshall Hall's conjecture.
Ismael Jimenez Calvo and G. Saez Moreno, Approximate Power roots in Z_m, Proceedings of ISC 2001 (Information Security); G. I. Davida and Y. Frankel, eds.; Springer, 2001; pp. 310-323.
I. Jiminez Calvo, J. Herranz, and G. Saez, A new algorithm to search for small nonzero |x^3-y^2| values, Math. Comp. 76 (268) (2009) 2435-2444.
L. V. Danilov, Diophantine equation x^3-y^2-k and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
L. V. Danilov, Letter to the editors, Mat. Zametki, 36:3 (1984), 457-458.
L. V. Danilov, Letter to the editor, Mathem. Notes, 36 (3) (1984), 726.
R. D'Mello, Marshall Hall's Conjecture and Gaps Between Integer Points on Mordell Elliptic Curves, arXiv preprint arXiv:1410.0078 [math.NT], 2014.
Noam D. Elkies, List of integers x,y with x<10^18, 0 < |x^3-y^2| < x^(1/2).
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

Cf. A179108, A179387, A200216.

For[x=1, True, x++, If[Abs[x^3-Round[Sqrt[x^3]]^2] < Sqrt[x] && !IntegerQ[Sqrt[x]], Print[x]]]

A200218 The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.

Original entry on oeis.org

-297, 548147655, -1019827620252441, 1897387247823873407415, -3530085179800800999132960777, 6567716416847133270037051381858983, -12219223258107727669457593220846745613305, 22733840433256343397153666138928891468676446359

Offset: 1

Artur Jasinski, Nov 14 2011

For x values see A200216.
For y values see A200217.
All terms in this sequence are of the form: 3^3 * 11(2^3 * 31 * 61^2 * k + 922807).

Cf. A078933, A179107, A179108, A179109, A179387, A179388, A199496.

aa = {}; uu = 682 + 61 * Sqrt[125]; Do[vv = Expand[uu^(2 * n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5 * tt^2 - 3000 * tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, dd], {n, 1, 10}]; aa
(* Recurrence generator of R. J. Mathar *)
dd = {-297, 548147655, -1019827620252441}; a0 = dd[[1]]; a1 = dd[[2]]; a2 = dd[[3]]; Do[a = a0 + 1860497 * a1 - 1860497 * a2; a0 = a1; a1 = a2; a2 = a; AppendTo[aa, a], {n, 1, 10}]; aa
(* Third one after Lucas numbers formula *)
Table[27/125 (-5 + (-1)^n ((-1)^(n + 1) 6 + LucasL[15 (-1 + 2 n)])), {n, 10}] (* Artur Jasinski, Nov 18 2011*)

3125 * a(n)^2 + 6750 * a(n) + 729 = 2916 * A200216(n).
a(n) = (A200216(n))^3 - (A200217(n))^2.
Conjecture: a(n) = -1860497 * a(n-1) + 1860497 * a(n-2) + a(n-3) with g.f. 297 * z * (1 + 14882 * z + z^2) / ( (z-1)*(z^2 + 1860498 * z+1) ). - R. J. Mathar, Nov 15 2011
Hyperelliptic curve (157464*y)^2 = (729 + 594*d + 125*d^2) (-729 + 13500*d + 15625*d^2)^2 is singular (has two cusps) and for this reason Danilov's sequence has infinitely many integer solutions. - Artur Jasinski, Nov 16 2011
a(n) = (27/125) * (-5 + (-1)^n * ((-1)^(n+1) * 6 + L(15*(2*n - 1)))) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

A200217 Danilov's sequence of y values satisfying 0 < |x^3 - y^2| < sqrt(x) with integer (x,y).

Original entry on oeis.org

28748141, 182720147509505842286585077, 1176722513851727970194784616032383058302343205, 7578123615032687003769196301877008424487234722410713810234126013

Offset: 1

Artur Jasinski, Nov 14 2011

nonn

See A078933 for further references.
All terms in this sequence are of the form 61*(2728*k + 2065).
Relations between y and x are given by the curve:
125*y^2 *(54 + 50*x^3 - 25*y^2) = (9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2.
Relations between y and d are given by the hyperelliptic curve:
(157464*y)^2 = (729 + 594*d + 125*d^2) (-729 + 13500*d + 15625*d^2)^2 is singular (has two cusps) and for these reasons Danilov's sequence has infinitely many integer solutions. - Artur Jasinski, Nov 16 2011

L. V. Danilov Letter to the editor, Math. Notes 36 (3) (1984) 726.

Cf. A078933, A200216 (x-values), A200218 (x^3-y^2), A179107 - A179109, A179387, A179388, A199496.

with(numtheory):
Di := 125 ;
cf := numtheory[cfrac](sqrt(Di),'periodic','quotients') ;
for i from 1 to 220 do
   x := nthnumer(cf,i) ;
   y := nthdenom(cf,i) ;
   rr := x^2-Di*y^2 ;
   if rr = -1 then
      t := x-5 ;
      if (t mod 5) = 2 then
              t := -t-10 ;
              y := -y ;
      end if;
      pk := t ;
      qk := y ;
      yM := qk*(pk^2+pk-1) ;
      yM := abs(yM) ;
      printf("%d,",yM) ;
   end if;
end do: # R. J. Mathar, Nov 15 2011

aa = {}; uu = 682 + 61*Sqrt[125]; Do[vv = Expand[uu^(2*n - 1)]; tt = ((-1)^n vv[[1]] + 57)/125; xx = (5^5*tt^2 - 3000*tt + 719); yy = Round[N[Sqrt[xx^3], 1000]]; dd = xx^3 - yy^2; AppendTo[aa, yy], {n, 1, 5}]; aa
(* recurrence formula of R. J. Mathar *)
dd = {28748141, 182720147509505842286585077, 1176722513851727970194784616032383058302343205, 7578123615032687003769196301877008424487234722410713810234126013, 48803313311937248954865638168364942372153001387358275397822506563724900540813098269, 314294607902465331119210305427552029679173295887697814635011836442516313036620521777783545650437642949}; a0 = dd[[1]]; a1 = dd[[2]]; a2 = dd[[3]]; a3 = dd[[4]]; a4 = dd[[5]]; a5 = dd[[6]]; Do[a = 6440022564929296994 a5 + 22291834190970757443015664937985 a4 + -41473935220466903245533179036528718020 a3 + 22291834190970757443015664937985 a2 + 6440022564929296994*a1 - a0; a0 = a1; a1 = a2; a2 = a3 = a4; a5 = a6; a6 = a; AppendTo[aa, a], {n, 1, 10}]; aa (* Artur Jasinski, Nov 15 2011 *)
CoefficientList[Series[-(61 (-1 + x) (471281 - 39648020168249880312376 x - 417898575330317669831476343067314 x^2 - 39648020168249880312376 x^3 + 471281 x^4))/((1 - 6440026026380244498 x + x^2) (1 - 1860498 x + x^2) (1 + 3461452808002 x + x^2)), {x, 0, 10}], x] (* Artur Jasinski, Nov 16 2011 *)
(* Lucas - Fibonacci formula *)
aa = {}; Do[If[n == 1, AppendTo[aa, 15 + 9 LucasL[15 (-1 + 2 n)] + 15 LucasL[30 (-1 + 2 n)] - 6 Fibonacci[15 (-1 + 2 n)] + Fibonacci[30 (-1 + 2 n)]], AppendTo[aa, 15/8 Fibonacci[15 (-1 + 2 n)] - 9/20 Fibonacci[30 (-1 + 2 n)] + 1/40 Fibonacci[45 (-1 + 2 n)]]], {n, 1, 10}]; aa (* Artur Jasinski, Nov 18 2011 *)

Conjecture: a(n) = +6440022564929296994*a(n-1) +22291834190970757443015664937985*a(n-2) -41473935220466903245533179036528718020*a(n-3) +22291834190970757443015664937985*a(n-4) +6440022564929296994*a(n-5) -a(n-6). - R. J. Mathar, Nov 15 2011
Equivalent conjecture g.f.: -61*(z-1) * (471281*z^4 -39648020168249880312376*z^3 -417898575330317669831476343067314*z^2 -39648020168249880312376*z +471281) / ( (z^2+3461452808002*z+1) *(z^2-6440026026380244498*z+1) *(z^2-1860498*z+1) ). - R. J. Mathar, Nov 15 2011
Formula by Lucas and Fibonacci numbers: a(1) = 15+9*L(15)+15*L(30)-6*F(15)+F(30), for n>1 a(n) = (15/8)*F(15(2n-1)) - (9/20)*F(30(2n-1)) + (1/40) * F(45(2n-1)) where F(k) is k-th Fibonacci number A000045(n) and L(k) is k-th Lucas number A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

A200656 Successive values x such that the Mordell elliptic curve x^3 - y^2 = d has extremal points with quadratic extension over the rationals.

Original entry on oeis.org

1942, 2878, 3862, 6100, 8380, 11512, 15448, 18694, 31228, 93844, 111382, 117118, 129910, 143950, 186145, 210025, 375376, 445528, 468472, 575800, 844596, 1002438, 1054062, 1193740, 1248412, 1326025, 1388545, 1501504, 1697908, 1782112, 1813660, 1873888, 1946737

Offset: 1

Artur Jasinski, Nov 20 2011

nonn

Definition: Extremal points on the Mordell elliptic curve x^3 - y^2 = d are points (x,y) such that x^3 - round(sqrt(x^3))^2 = d. For values d for successive x independent of the extensions see A077119.
For y values see A200657.
For d values see A200658.
Definition: Secondary terms occur when there exist integers k such that A200656 is divisible by k^2, A200657 is divisible by k^3 and A200658 is divisible by k^6.
Terms free of such k are primary terms; see A201047. Secondary terms are, e.g., a(6)=a(2)*2^2, a(7)=a(3)*2^2, a(17)=a(10)*2^2, a(18)=a(11)*2^2, a(19)=a(12)*2^2, a(21)=a(10)*3^2.
For successive secondary terms, see A201048.
A200216 is a subsequence of this sequence.

Peter J. C. Moses and Artur Jasinski, Table of n, a(n) for n = 1..185 (all terms below 10^9)

Cf. A200216, A201047, A201048.

Cf. A077119, A200657, A200658.

A134105 Complete list of solutions to y^2 = x^3 + 297; sequence gives x values.

Original entry on oeis.org

-6, -2, 3, 4, 12, 34, 48, 1362, 93844

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134104.

The parameter -297 of the curve corresponds to A200218(1). a(9)=A200216(1). - Artur Jasinski, Nov 29 2011

Cf. A134104, A134103, A134107, A029728, A134042, A134074.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029728 */

sol[x_] := Solve[y > 0 && x^3 - y^2 == -297, y, Integers];
Reap[For[x = 1, x < 10^5, x++, sx = sol[x]; If[sx != {}, xy = {x, y} /. sx[[1]]; Print[xy]; Sow[xy]]; sx = sol[-x]; If[sx != {}, xy = {-x, y} /. sx[[1]]; Print[xy]; Sow[xy]]]][[2, 1]][[All, 1]] // Sort (* Jean-François Alcover, Feb 07 2020 *)

[i[0] for i in EllipticCurve([0, 297]).integral_points()] # Seiichi Manyama, Aug 26 2019

A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

Original entry on oeis.org

10, 2, 2, 5, 5, 130, 185, 5, 2, 2, 10, 10, 5, 5, 10, 17, 17, 5, 5, 5, 53, 53, 13, 13, 1490, 5, 2, 2, 5, 1565, 5

Offset: 1

Artur Jasinski, Nov 29 2011

Conjecture (Jasiński): The numbers in this sequence are multiplicative combinations of: primes congruent to 1 or 2 modulo 4 (A002313), Pythagorean primes (A002144), the number 2, and norms of Gaussian primes A055025.

Cf. A200216, A200656, A201047.

Minor edits by N. J. A. Sloane, Feb 23 2014

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

Artur Jasinski, Nov 25 2011

nonn

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.

			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

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

Cf. A001333, A200216, A200656.

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

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 *)

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

{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 */

{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 */

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

Artur Jasinski, Nov 25 2011

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.

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).

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

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

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

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

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

A201225 Values x for infinite sequence x^3-y^2 = d with decreasing coefficient r=sqrt(x)/d which tend to 1/(1350*sqrt(5))or infinity family of solutions Mordell curve with extension sqrt(5).

Original entry on oeis.org

6100, 2305180, 748476100, 241118603980, 77641444770100, 25000340035616380, 8050032494909496100, 2592085474592828222380, 834643472994047002110100, 268752606222334691877221980, 86537504560185639786707316100, 27864807715774753485364243735180

Offset: 1

Artur Jasinski, Nov 28 2011

nonn

a(1) = A200656(4) = A201047(4).
a(2) = A200656(36) = A201047(26).
All points in this sequence are extremal points (definition see A200656) and from these reason is subset of A200656 and primary (definition see A200656) and from these reason is subset of A201047.

Index entries for linear recurrences with constant coefficients, signature (341,-6138,6138,-341,1).

Cf. A200216, A200656.

LinearRecurrence[{341,-6138,6138,-341,1},{6100,2305180,748476100,241118603980,77641444770100},20] (* Harvey P. Dale, Aug 17 2016 *)

G.f.: (20*(-305-11254*z+7424*z^2-346*z^3+z^4))/((-1+z)*(1- 322*z+z^2)*(1-18*z+z^2)).

a(n) = 341*a(n-1) - 6138*a(n-2) + 6138*a(n-3) - 341*a(n-4) + a(n-5).

cp's OEIS Frontend

A200335 a(n) = sqrt((A200216(n)+1)/5).

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A078933 Good examples of Hall's conjecture: integers x such that 0 < |x^3 - y^2| < sqrt(x) for some integer y.

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A200218 The differences x^3 - y^2 of Danilov's subsequence of good Hall's examples A078933.

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A200217 Danilov's sequence of y values satisfying 0 < |x^3 - y^2| < sqrt(x) with integer (x,y).

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A200656 Successive values x such that the Mordell elliptic curve x^3 - y^2 = d has extremal points with quadratic extension over the rationals.

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A134105 Complete list of solutions to y^2 = x^3 + 297; sequence gives x values.

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A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

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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.

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