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

28748141, 182720147509505842286585077, 1176722513851727970194784616032383058302343205, 7578123615032687003769196301877008424487234722410713810234126013

Offset: 1

Author

Artur Jasinski, Nov 14 2011

Keywords

• nonn

Comments

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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