A193682 Period 8: repeat [0, 1, 2, 3, 0, 3, 2, 1].

0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 1, 0, 1, 2, 3, 0

Offset: 0

Author

Wolfdieter Lang, Sep 30 2011

Keywords

• nonn
• easy

Comments

This sequence can be continued periodically for negative values of n.

See a comment on A203571 where a k-family of 2k-periodic sequences P_k has been defined. The present sequence is P_4. - Wolfdieter Lang, Feb 02 2012

Examples

			a(10) = 10(mod 4) = 2 because 10\4 = floor(10/4)=2 is even; the parity is +1.
a(7) = (4-7)(mod 4) = 1 because 7\4 = floor(7/4)=1 is odd; the parity is -1.
		

Links

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

Crossrefs

Cf. A193680 (mod 3 case).

Cf: A203571.

Programs

Magma

&cat [[0, 1, 2, 3, 0, 3, 2, 1]^^15]; // Vincenzo Librandi, Oct 17 2018

Mathematica

PadRight[{}, 120, {0, 1, 2, 3, 0, 3, 2, 1}] (* Vincenzo Librandi, Oct 17 2018 *)

PARI

a(n)=[0, 1, 2, 3, 0, 3, 2, 1][n%8+1] \\ Charles R Greathouse IV, Oct 16 2015

Python

def A193682(n): return (0,1,2,3,0,3,2,1)[n&7] # Chai Wah Wu, Jan 26 2023

Formula

a(n) = n mod 4 if (-1)^floor(n/4)=+1, otherwise (4-n) mod 4, n >= 0. (-1)^floor(n/4) is the parity of the quotient floor(n/4). This quotient is sometimes denoted by n\4.

O.g.f.: x*(1+2*x+3*x^2+3*x^4+2*x^5+x^6)/( (1-x)*(1+x)*(1+x^2)*(1+x^4)).

a(n) = floor(410107/33333333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013

a(n) = floor(2323/21845*4^(n+1)) mod 4. - Hieronymus Fischer, Jan 04 2013