A308778 Central element(s) in the period of the continued fraction expansion of sqrt(n), or 0 if no such element exists, or -1 if n is a square.

-1, -1, 0, 1, -1, 0, 2, 1, 1, -1, 0, 3, 2, 1, 2, 1, -1, 0, 4, 3, 2, 2, 4, 3, 1, -1, 0, 5, 2, 1, 2, 5, 1, 2, 4, 1, -1, 0, 6, 4, 3, 2, 2, 5, 2, 2, 6, 5, 1, -1, 0, 7, 2, 1, 6, 2, 2, 4, 1, 7, 2, 2, 6, 1, -1, 0, 8, 7, 4, 4, 2, 7, 2, 5, 1, 1, 4, 2, 4, 7, 1, -1, 0

Offset

0,7

Comments

The continued fraction expansion of sqrt(n) is periodic (where n is no square), and the period splits in two halves which are mirrored around the center. With r = floor(sqrt(n)) the expansion takes one of the forms:

[r; i, j, k, ..., m, m, ..., k, j, i, 2*r] (odd period length) or

[r; i, j, k, ..., m, ..., k, j, i, 2*r] (even period length)

[r; 2*r] (empty symmetric part, for n = r^2 + 1)

This sequence lists the central element(s) m, or 0 for n = r^2 + 1, or -1 for n = r^2.

a(k^2-1) = 1 for k >= 2. - Robert Israel, Nov 04 2019

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Georg Fischer, Table of the continued fractions of sqrt(0..20000)

Oskar Perron, Die Lehre von den Kettenbrüchen, B. G. Teubner (1913), section 24, p. 87 ff.

Example

CF(sqrt(2906)) = [53;1,9,1,3,1,3,1,1,14,1,5,2,2,5,1,14,1,1,3,1,3,1,9,1,106], odd period, two central elements, a(2906) = 2.

Maple

f:= proc(n) local L, m;
  if issqr(n) then return -1
  elif issqr(n-1) then return 0
  fi;
  L:= numtheory:-cfrac(sqrt(n), periodic, quotients);
  m:= nops(L[2]);
  L[2][floor(m/2)]
end proc:
map(f, [$0..100]); # Robert Israel, Nov 04 2019

Mathematica

Array[Which[IntegerQ@ Sqrt@ #, -1, IntegerQ@ Sqrt[# - 1], 0, True, #[[Floor[Length[#]/2]]] &@ Last@ ContinuedFraction@ Sqrt@ #] &, 83, 0] (* Michael De Vlieger, Jul 07 2019 *)

Crossrefs

Cf. A031509-A031688.

Sequence in context: A336931 A363953 A182662 * A127284 A120691 A111941

Adjacent sequences: A308775 A308776 A308777 * A308779 A308780 A308781

Keyword

sign,look

Author

Georg Fischer, Jun 24 2019

Status

approved