A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Author

Jaume Oliver Lafont, Oct 15 2009

Keywords

• nonn
• mult
• easy

Examples

			G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for characteristic functions
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Crossrefs

Characteristic function of A042968, whose complement A008586 gives the positions of zeros (after its initial term).

Absolute values of A046978, A075553, A131729, A358839, and for n >= 1, also of A112299 and of A257196.

Sequence A152822 shifted by two terms.

Parity of A327860, A331733, A342002, A351083 and A345000.

Row 3 of A225145, Column 2 of A229940 (after the initial term).

First differences of A057353. Sum of A359370 and A359372.

Cf. A000035, A011655, A011558, A097325, A109720, A168181, A168182, A168184, A145568, A168185 (characteristic functions for numbers that are not multiples of k = 2, 3 and 5..12).

Cf. A016628, A164985, A165132.

Cf. A010873, A033436, A069733 (inverse Möbius transform), A121262 (one's complement), A190621 [= n*a(n)], A355689 (Dirichlet inverse).

Programs

Magma

[Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

Maple

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

Mathematica

PadRight[{},120,{0,1,1,1}] (* Harvey P. Dale, Jul 04 2013 *)
Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)
a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PARI

{a(n) = !!(n%4)};

Python

def A166486(n): return (0,1,1,1)[n&3] # Chai Wah Wu, Jan 03 2023

Formula

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

a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).

Sum_{k>0} a(k)/(k*3^k) = log(5)/4.

From Reinhard Zumkeller, Nov 30 2009: (Start)

Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0.

a(n) = 1-A121262(n).

a(A042968(n))=1; a(A008586(n))=0.

A033436(n) = Sum{k=0..n} a(k)*(n-k). (End)

a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010

a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010

Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011

Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011

a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011

a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13 2012

For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014

a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015

For n >= 1, a(n) = A053866(A225546(n)) = A000035(A331733(n)). - Antti Karttunen, Jul 07 2020

a(n) = signum(n mod 4). - Alois P. Heinz, May 12 2021

From Antti Karttunen, Dec 28 2022: (Start)

a(n) = [A010873(n) > 0], where [ ] is the Iverson bracket.

a(n) = abs(A046978(n)) = abs(A075553(n)) = abs(A131729(n)) = abs(A358839(n)).

For all n >= 1, a(n) = abs(A112299(n)) = abs(A257196(n))

a(n) = A152822(2+n).

a(n) = A359370(n) + A359372(n). (End)

E.g.f.: (cosh(x) - cos(x))/2 + sinh(x). - Stefano Spezia, Aug 04 2025

Extensions

Secondary definition (from Reinhard Zumkeller's Nov 30 2009 comment) added to the name by Antti Karttunen, Dec 20 2022