A031688 -id:A031688 - cp's OEIS frontend

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

Georg Fischer, Jun 24 2019

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

			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.

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.

Cf. A031509-A031688.

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

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

A031599 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 11.

Original entry on oeis.org

809, 2581, 2785, 2890, 4217, 5353, 6949, 7114, 7621, 9698, 9893, 10289, 11177, 11597, 12461, 15325, 15817, 17429, 18218, 18754, 19298, 19946, 20506, 20789, 21361, 22234, 23833, 25693, 27085, 27737, 29065, 29741, 30202, 30425, 30545, 30890, 31237, 32290

Offset: 1

David W. Wilson

nonn

cf11Q[n_]:=Module[{s=Sqrt[n],len,cf},cf=If[IntegerQ[s],{2,2}, ContinuedFraction[ s][[2]]];len=Length[cf];OddQ[len]&&cf[[(len+1)/2]] == 11]; Select[Range[32000],cf11Q] (* Harvey P. Dale, Mar 18 2018 *)

Corrected and extended by Harvey P. Dale, Mar 18 2018

Definitions of A031599-A031688 clarified by N. J. A. Sloane, Aug 17 2021

cp's OEIS Frontend

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.

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