A132429 Period 4: repeat [3, 1, -1, -3].

3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1

Offset: 0

Author

Paul Curtz, Nov 13 2007

Keywords

• sign
• easy

Comments

Nonsimple continued fraction expansion of (7 + 3*sqrt(5))/2 = 6.85410196624... = 1 + A090550. - R. J. Mathar, Mar 08 2012

Pisano period lengths: 1, 1, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... . - R. J. Mathar, Aug 10 2012

Links

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

Crossrefs

Cf. A084101 (1, 3, 3, 1), A090550.

Programs

Haskell

a132429 = (3 -) . (* 2) . (`mod` 4)
a132429_list = cycle [3, 1, -1, -3]  -- Reinhard Zumkeller, Aug 15 2015

Magma

&cat [[3, 1, -1, -3]^^30]; // Wesley Ivan Hurt, Jul 10 2016

Maple

A132429:=n->3 - 2 * (n mod 4); seq(A132429(n), n=0..100); # Wesley Ivan Hurt, Apr 18 2014

Mathematica

PadRight[{}, 104, {3,1,-1,-3}] (* Harvey P. Dale, Nov 12 2011 *)

PARI

a(n)=3-2*(n%4) \\ Jaume Oliver Lafont, Aug 28 2009

Python

def A132429(n): return 3 - 2*(n & 3) # Chai Wah Wu, May 25 2022

Formula

G.f.: (3 + 4*x + 3*x^2)/((1+x)*(1+x^2)). - Jaume Oliver Lafont, Aug 30 2009

a(n) = (-1)^n + 2(-1)^((2n + (-1)^n - 1)/4). - Brad Clardy, Mar 10 2013

a(n) = 3 - 2*(n mod 4). - Joerg Arndt, Mar 10 2013

a(n) = (-1)^n + 2(-1)^floor(n/2). - Wesley Ivan Hurt, Apr 17 2014

From Wesley Ivan Hurt, Jul 10 2016: (Start)

a(n) + a(n-1) + a(n-2) + a(n-3) = 0 for n>2, a(n) = a(n-4) for n>3.

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