A199496 -id:A199496 - cp's OEIS frontend

A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

93844, 322001299796379844, 1114592308630995805123571151844, 3858108676488182444301031186675778188809844, 13354661111806898918013326915229994453818137920195953844

Offset: 1

Artur Jasinski, Nov 14 2011

nonn

For y values see A200217.
For x^3-y^2 values see A200218.
The values (a(n)+1)/5 are perfect squares: for sqrt((a(n)+1)/5) see A200335.
This sequence is an infinite subset of A078933. - Artur Jasinski, Nov 27 2011

			|93844^3 - (round(sqrt(93844^3)))^2| < sqrt(93844).

For references see A078933 and A179386.

L. V. Danilov, Letter to the Editor, Mat. Zametki, 1984, Volume 36, Issue 3, Pages 457-458.
L. V. Danilov, Letter to the editor, Math. 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.

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

Cf. A200217, A200218, A200335.

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, xx], {n, 1, 6}]; aa
(* second program follows the generator formula *)
data = Table[(7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2 n - 1)) - (1/f)^(15 (2 n - 1))) + (1/20)*(f^(30 (2 n - 1)) + (1/f)^(30 (2 n - 1)))) /. f -> GoldenRatio, {n, 1, 6}]; data // FunctionExpand // ExpandAll // Simplify (* Bob Hanlon (hanlonr(AT)cox.net) *)
(* third program uses the Lucas numbers formula *)
Table[7/10 + (-1)^(n + 1) 3/5 LucasL[15*(2 n - 1)] +
  1/20 LucasL[30*(2 n - 1)] , {n, 1, 10}] (* Artur Jasinski, Nov 18 2011 *)

u = quadunit(20)^5
for(i=1,6, v = u^(2*i-1); t = ((-1)^i * real(v) + 57) / 125; print(5^5*t^2 - 3000*t + 719) ) \\ Noam D. Elkies

from sympy import lucas
def A200216(n): return (14+12*(lucas(k:=30*n-15) if n&1 else -lucas(k:=30*n-15))+lucas(k<<1))//20 # Chai Wah Wu, Jun 19 2024

Conjecture: a(n) = 3461450947505*a(n-1) + 6440022564931157490*a(n-2) - 6440022564931157490*a(n-3) - 3461450947505*a(n-4) + a(n-5). - R. J. Mathar, Nov 15 2011
Conjecture: g.f. 4092*(1-z)/(5*(1+1860498*z+z^2)) - 7/(10*(z-1)) + 930249*(1-z)/(10*(1-3461452808002*z+z^2)). - R. J. Mathar, Nov 15 2011
3125*A200218(n)^2 + 6750*A200218(n) + 729 = 2916*a(n). - Artur Jasinski, Nov 15 2011
125*y^2 *(54 + 50*x^3 - 25*y^2)=(9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2. - Artur Jasinski, Nov 16 2011
a(n) = 7/10 - (6/5)*(-1)^n*(1/2)*(f^(15*(2*n-1))-(1/f)^(15*(2*n-1))) + (1/20)*(f^(30*(2*n-1))+(1/f)^(30*(2*n-1))), where f is golden ratio constant = (1+sqrt(5)/2). - Artur Jasinski, Nov 17 2011
a(n) = 7/10 + (3/5)*L(15*(2*n - 1))*(-1)^(n+1) + (1/20)*L(30*(2*n - 1)) where L(k) is the k-th Lucas number: A000204(n) or A000032(n+1). - Artur Jasinski, Nov 18 2011

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

A199497 Distances d=x^3-y^2 for positive good examples of Hall's conjecture: positive integers d < sqrt(x).

Original entry on oeis.org

207, 307, 852135, 2767769, 5190544, 1641843, 105077952, 30032270, 548147655, 1651035656, 1878790553, 4101044247, 13027681441

Offset: 1

Artur Jasinski, Nov 07 2011

For values x see A199496.

For values y see A199498.

Cf. A078933, A199496, A199498.

A199498 Values y for positive good examples of Hall's conjecture: positive integers d = x^3-y^2 < sqrt(x).

Original entry on oeis.org

223063347, 911054064, 929989991784733049, 43868513629203032816, 211477180624706647625, 447884928428402042307918, 3583079427427216338463344, 7441505802879036345061579, 182720147509505842286585077, 31012309752051601656131750312, 277382747164996776244709473092

Offset: 1

Artur Jasinski, Nov 07 2011

nonn

For values x see A199496.

For values d see A199497.

Cf. A078933, A199496, A199497.

A273927 Absolute difference between A000290(n) and the nearest term of A000578.

Original entry on oeis.org

1, 3, 1, 8, 2, 9, 15, 37, 17, 25, 4, 19, 44, 20, 9, 40, 54, 19, 18, 57, 71, 28, 17, 64, 104, 53, 217, 55, 112, 100, 39, 24, 89, 156, 106, 35, 38, 113, 190, 128, 47, 36, 121, 208, 172, 81, 12, 107, 204, 244, 143, 40, 65, 172, 281, 239, 126, 11, 106, 225, 346

Offset: 1

Felix Fröhlich, Jun 04 2016

nonn

Marshall Hall, Jr. conjectured that there exists a constant c > 0 such that for any integers x and y with y^2 != x^3 the relation abs(y^2 - x^3) > c*sqrt(abs(x)) holds.

Wikipedia, Hall's conjecture.

Cf. A078933, A199496, A199497, A199498.

nearestcube(n) = my(x=n-1, y=n+1); while(!ispower(x, 3) && !ispower(y, 3), x--; y++); if(ispower(x, 3), return(x)); if(ispower(y, 3), return(y))
a(n) = abs(n^2 - nearestcube(n^2))

cp's OEIS Frontend

A200216 Danilov's sequence of x satisfying 0 < |x^3-y^2| < sqrt(x) with integer (x,y).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A199497 Distances d=x^3-y^2 for positive good examples of Hall's conjecture: positive integers d < sqrt(x).

Views

Author

Keywords

Comments

Crossrefs

A199498 Values y for positive good examples of Hall's conjecture: positive integers d = x^3-y^2 < sqrt(x).

Views

Author

Keywords

Comments

Crossrefs

A273927 Absolute difference between A000290(n) and the nearest term of A000578.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI