A146326 Length of the period of the continued fraction of (1+sqrt(n))/2.

0, 2, 2, 0, 1, 4, 4, 4, 0, 2, 2, 2, 1, 4, 2, 0, 3, 6, 6, 4, 2, 6, 4, 4, 0, 2, 2, 4, 1, 2, 8, 4, 4, 4, 2, 0, 3, 6, 6, 8, 5, 4, 10, 6, 2, 8, 4, 4, 0, 2, 2, 4, 1, 6, 4, 2, 6, 6, 6, 4, 3, 4, 2, 0, 3, 6, 10, 6, 4, 6, 8, 4, 9, 6, 4, 8, 2, 4, 4, 4, 0, 2, 2, 2, 1, 6, 2, 8, 7, 2, 8, 8, 2, 12, 4, 8, 9, 4, 2, 0

Offset: 1

Author

Artur Jasinski, Oct 30 2008

Keywords

• nonn

Comments

First occurrence of n in this sequence see A146343.

Records see A146344.

Indices where records occurred see A146345.

a(n) =0 for n = k^2 (A000290).

a(n) =1 for n = 4 k^2 + 4 k + 5 (A078370). For primes see A005473.

a(n) =2 for n in A146327. For primes see A056899.

a(n) =3 for n in A146328. For primes see A146348.

a(n) =4 for n in A146329. For primes see A028871 - {2}.

a(n) =5 for n in A146330. For primes see A146350.

a(n) =6 for n in A146331. For primes see A146351.

a(n) =7 for n in A146332. For primes see A146352.

a(n) =8 for n in A146333. For primes see A146353.

a(n) =9 for n in A143577. For primes see A146354.

a(n)=10 for n in A146334. For primes see A146355.

a(n)=11 for n in A146335. For primes see A146356.

a(n)=12 for n in A146336. For primes see A146357.

a(n)=13 for n in A333640. For primes see A146358.

a(n)=14 for n in A146337. For primes see A146359.

a(n)=15 for n in A146338. For primes see A146360.

a(n)=16 for n in A146339. For primes see A146361.

a(n)=17 for n in A146340. For primes see A146362.

Examples

			a(2) = 2 because continued fraction of (1+sqrt(2))/2 = 1, 4, 1, 4, 1, 4, 1, ... has period (1,4) length 2.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..20000.

Crossrefs

Cf. A003285, A146343, A146344, A146345.

Cf. also A000290, A078370, A146327-A146342; A005473, A056899, A146348-A146362.

Programs

Maple

A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: seq(A146326(n),n=1..100) ; # R. J. Mathar, Sep 06 2009

Mathematica

Table[cf = ContinuedFraction[(1 + Sqrt[n])/2]; If[Head[cf[[-1]]] === List, Length[cf[[-1]]], 0], {n, 100}]
f[n_] := Length@ ContinuedFraction[(1 + Sqrt[n])/2][[-1]]; Array[f, 100] (* Robert G. Wilson v, Apr 11 2017 *)

Formula

a(n) = 0 iff n is a square (A000290). - Robert G. Wilson v, Apr 11 2017

Extensions

a(39) and a(68) corrected by R. J. Mathar, Sep 06 2009