A309666 a(n) is the least k such that the denominators of continued fraction convergents for sqrt(k) match the first n Fibonacci numbers.

2, 3, 7, 7, 13, 58, 58, 135, 819, 819, 2081, 13834, 13834, 35955, 244647, 244647, 639389, 4374866, 4374866, 11448871, 78439683, 78439683, 205337953, 1407271538, 1407271538, 3684200835, 25251313255, 25251313255, 66108441037, 453111560266, 453111560266, 1186259960295, 8130736409715, 8130736409715, 21286537898177

Offset

1,1

Comments

Aside from the first term, this appears to be a subset of A060215.

Same as A071296 if you drop a(0) and replace each repeated pair x,x with 0,x (credit to Daniel Suteu for pointing this out).

These are also the least a(n) such that the continued fraction expansion for sqrt(a(n) - floor(a(n))) begins with (n-1) 1's.

Links

Table of n, a(n) for n=1..35.

Formula

Conjectures from Colin Barker, Aug 26 2019: (Start)

G.f.: x*(2 + x + 4*x^2 - 42*x^3 - 15*x^4 - 39*x^5 + 100*x^6 + x^7 - 61*x^8 + 172*x^9 + 31*x^10 - 17*x^11 + 26*x^12 - 2*x^13 + x^14 - 2*x^15) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x + x^2)*(1 - x - x^2)*(1 + x + 2*x^2 - x^3 + x^4)*(1 + 3*x + 8*x^2 + 3*x^3 + x^4)).

a(n) = a(n-1) + 21*a(n-3) - 21*a(n-4) - 50*a(n-6) + 50*a(n-7) - 86*a(n-9) + 86*a(n-10) - 13*a(n-12) + 13*a(n-13) + a(n-15) - a(n-16) for n>16.

(End)

Example

For n = 5 the convergents of sqrt(13) are 3/1, 4/1, 7/2, 11/3, 18/5, 119/33, ... and the first five denominators are 1, 1, 2, 3, 5, which match the first five Fibonacci numbers. Since 13 is the first number with this property, then a(5) = 13.

Mathematica

c = 1;
n = 2;
F = Table[Fibonacci[n], {n, 20}];
While[c <= 14,
If[! IntegerQ[Sqrt[n]]
   &&
   Denominator[Convergents[Sqrt[n], c]] == F[[1 ;; c]],
  Print[n, "  ", Denominator[Convergents[Sqrt[n], c]]];
  c++; n--];
n++
]

Crossrefs

Cf. A060215, A071296.

Sequence in context: A276730 A179894 A060215 * A085420 A027672 A322138

Adjacent sequences: A309663 A309664 A309665 * A309667 A309668 A309669

Keyword

nonn

Author

Greg Dresden, Aug 11 2019

Status

approved