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

%I A200217 #78 Jul 25 2024 06:53:44
%S A200217 28748141,182720147509505842286585077,
%T A200217 1176722513851727970194784616032383058302343205,
%U A200217 7578123615032687003769196301877008424487234722410713810234126013
%N A200217 Danilov's sequence of y values satisfying 0 < |x^3 - y^2| < sqrt(x) with integer (x,y).
%C A200217 See A078933 for further references.
%C A200217 All terms in this sequence are of the form 61*(2728*k + 2065).
%C A200217 Relations between y and x are given by the curve:
%C A200217 125*y^2 *(54 + 50*x^3 - 25*y^2) = (9 - 6*x + 5*x^2)*(-9 + 15*x + 25*x^2)^2.
%C A200217 Relations between y and d are given by the hyperelliptic curve:
%C A200217 (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
%H A200217 L. V. Danilov <a href="http://dx.doi.org/10.1007/BF01141949">Letter to the editor</a>, Math. Notes 36 (3) (1984) 726.
%F A200217 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
%F A200217 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
%F A200217 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
%p A200217 with(numtheory):
%p A200217 Di := 125 ;
%p A200217 cf := numtheory[cfrac](sqrt(Di),'periodic','quotients') ;
%p A200217 for i from 1 to 220 do
%p A200217    x := nthnumer(cf,i) ;
%p A200217    y := nthdenom(cf,i) ;
%p A200217    rr := x^2-Di*y^2 ;
%p A200217    if rr = -1 then
%p A200217       t := x-5 ;
%p A200217       if (t mod 5) = 2 then
%p A200217               t := -t-10 ;
%p A200217               y := -y ;
%p A200217       end if;
%p A200217       pk := t ;
%p A200217       qk := y ;
%p A200217       yM := qk*(pk^2+pk-1) ;
%p A200217       yM := abs(yM) ;
%p A200217       printf("%d,",yM) ;
%p A200217    end if;
%p A200217 end do: # _R. J. Mathar_, Nov 15 2011
%t A200217 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
%t A200217 (* recurrence formula of _R. J. Mathar_ *)
%t A200217 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 *)
%t A200217 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 *)
%t A200217 (* Lucas - Fibonacci formula *)
%t A200217 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 *)
%Y A200217 Cf. A078933, A200216 (x-values), A200218 (x^3-y^2), A179107 - A179109, A179387, A179388, A199496.
%K A200217 nonn
%O A200217 1,1
%A A200217 _Artur Jasinski_, Nov 14 2011