A013943 Period of continued fraction for sqrt(m), m = n-th nonsquare.

1, 2, 1, 2, 4, 2, 1, 2, 2, 5, 4, 2, 1, 2, 6, 2, 6, 6, 4, 2, 1, 2, 4, 5, 2, 8, 4, 4, 4, 2, 1, 2, 2, 2, 3, 2, 10, 8, 6, 12, 4, 2, 1, 2, 6, 5, 6, 4, 2, 6, 7, 6, 4, 11, 4, 2, 1, 2, 10, 2, 8, 6, 8, 2, 7, 5, 4, 12, 6, 4, 4, 2, 1, 2, 2, 5, 10, 2, 6, 5, 2, 8, 8, 10, 16, 4, 4, 11, 4, 2, 1, 2, 12, 2, 2, 9, 6, 8, 15, 2, 6, 6

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
• Kival Ngaokrajang, Illustration of initial terms, periodic are colored.
• Eric Weisstein's World of Mathematics, Continued Fraction.

Crossrefs

Cf. A003285, A035015.

Programs

Mathematica

nonSquares = Select[Range[120], !IntegerQ[Sqrt[#]]&]; a[n_] := Length[ Last[ ContinuedFraction[ Sqrt[ nonSquares[[n]] ]]]]; Table[a[n], {n, 1, Length[nonSquares]}] (* Jean-François Alcover, May 27 2013 *)

Python

from math import isqrt
from sympy.ntheory.continued_fraction import continued_fraction_periodic
def A013943(n): return len(continued_fraction_periodic(0,1,n+(k:=isqrt(n))+int(n>=k*(k+1)+1))[-1]) # Chai Wah Wu, Jul 20 2024