A146325 Period 3: repeat [1, 4, 1].

1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1

Offset: 1

Author

Artur Jasinski, Oct 30 2008

Keywords

• nonn
• easy

Comments

Continued fraction of (1 + sqrt(26))/5 = A188659.

Digital roots of the centered triangular numbers A005448. - Ant King, May 08 2012

Also the digital roots of centered 12-gonal numbers A003154. - Peter M. Chema, Dec 20 2023

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A003154, A005448, A021337, A131534 (square roots), A188659.

Programs

Magma

&cat [[1,4,1]^^40]; // Bruno Berselli, Jun 27 2016

Maple

seq(op([1, 4, 1]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016

Mathematica

Table[Round[N[4 (Cos[(2 n - 1) ArcTan[Sqrt[3]]])^2, 100]], {n, 1, 100}]
PadLeft[{},111,{1,4,1}] (* Harvey P. Dale, Sep 18 2011 *)

PARI

a(n)=1+3*(n%3==2) \\ Jaume Oliver Lafont, Mar 24 2009

Formula

a(n) = 4*(cos((2*n - 1)*Pi/3))^2 = 4 - 4*(sin((2*n - 1)*Pi/3))^2.

a(n+3) = a(n).

a(n) = 2 - cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3).

O.g.f.: x*(1+4*x+x^2)/(1-x^3). [Richard Choulet, Nov 03 2008]

a(n) = 6 - a(n-1) - a(n-2) for n>2. - Ant King, Jun 12 2012

a(n) = (n mod 3)^(n mod 3). - Bruno Berselli, Jun 27 2016

a(n) = 1 + A021337(n) for n>0. - Wesley Ivan Hurt, Jul 01 2016