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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. 2
-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 (list; graph; refs; listen; history; text; internal format)
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 Kettenbr├╝chen, 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: A249303 A319081 A182662 * A127284 A120691 A111941

Adjacent sequences:  A308775 A308776 A308777 * A308779 A308780 A308781

KEYWORD

sign,look

AUTHOR

Georg Fischer, Jun 24 2019

STATUS

approved

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Last modified December 9 17:18 EST 2019. Contains 329879 sequences. (Running on oeis4.)