A068073 Period 4 sequence [ 1, 2, 3, 2, ...].

1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1

Offset: 0

Author

Robert G. Wilson v, Mar 01 2002

Keywords

• easy
• nonn

Comments

Continued fraction expansion of (2+sqrt(14))/4. - Klaus Brockhaus, May 01 2010

The sequence is like a sawtooth wave of period 4. - Michael Somos, Feb 13 2011

Examples

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + x^8 + 2*x^9 + ...
		

Links

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

Crossrefs

Cf. A000034, A000120, A001045, A028356, A164356.

Cf. A177033 (decimal expansion of (2+sqrt(14))/4). - Klaus Brockhaus, May 01 2010

Programs

Mathematica

CoefficientList[ Series[(1 + 2x + 3x^2 + 2x^3)/(1 - x^4), {x, 0, 85}], x]
a[ n_] := {2, 3, 2, 1}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 17 2015 *)
PadRight[{},120,{1,2,3,2}] (* Harvey P. Dale, Jun 13 2020 *)

PARI

{a(n) = [1, 2, 3, 2] [n%4 + 1]}; /* Michael Somos, Feb 13 2011 */

PARI

{a(n) = n%4 + 1 - 2 * (n%4 == 3)}; /* Michael Somos, Feb 13 2011 */

PARI

{a(n) = 2 + kronecker( -4, n-1)}; /* Michael Somos, Feb 13 2011 */

Formula

G.f.: (1 + 2*x + 3*x^2 + 2*x^3) / (1 - x^4).

Conjecture: a(n) = Sum_{k=0..n} e^(i*Pi*(A000120(A001045(n)) - A001045(A000120(n)))), i=sqrt(-1). - Paul Barry, Jan 14 2005

From Paul Barry, Jan 14 2005: (Start)

G.f.: (1 + x + 2x^2)/(1 - x + x^2 - x^3);

a(n) = 2 - cos(Pi*n/2). (End)

Moebius transform is length 4 sequence [2, 1, 0, -2]. - Michael Somos, Feb 13 2011

a(n) = 2 - A056594(n). - Bruno Berselli, Mar 10 2011

a(n) = a(-n) = a(n+4) for all n in Z. - Michael Somos, Apr 17 2015

2 * a(n) = A164356(n) unless n=0. - Michael Somos, Apr 17 2015

G.f.: 1 / (1 - 2*x / (1 + x / (2 - 5*x / (1 + 16*x / (5 - x))))). - Michael Somos, Jan 20 2017

G.f.: 2 / (1 - x) - 1 / (1 + x^2). - Michael Somos, Jan 07 2019

a(n) = abs(((n+2) mod 4)-2) + 1. - Daniel Jiménez, Jan 14 2023