A179850 Characteristic function of numbers that are congruent to {0, 1, 3, 4} mod 5.

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

Offset: 0

Author

Michael Somos, Jan 10 2011

Keywords

• nonn

Comments

a(n) is also the characteristic sequence for the mod m reduced odd numbers (i.e., gcd(2*n+1,m)=1, n>=0) for each modulus m from 5*A003592 = [5, 10, 20, 25, 40, 50, 80, 100, 125,...]. [Wolfdieter Lang, Feb 04 2012]

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^8 + x^9 + x^10 + x^11 + x^13 + ...
G.f. = q + q^3 + q^7 + q^9 + q^11 + q^13 + q^17 + q^19 + q^21 + q^23 + ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..4999
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Crossrefs

Cf. A003592, A047207, A011558, A130782.

Programs

Mathematica

a[ n_] := Sign @ Mod[n - 2, 5]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := {1, 0, 1, 1, 1}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 17 2015 *)

PARI

{a(n) = sign( (n - 2) % 5 )};

PARI

{a(n) = [1, 1, 0, 1, 1][n%5 + 1]};

Formula

a(n) = b(2*n + 1) where b(n) is completely multiplicative with b(2) = b(5) = 0, otherwise b(p) = 1.

Coefficient of q^(2*n + 1) in q * (1 - q^4) * (1 - q^12) / ((1 - q^2) * (1 - q^6) * (1 - q^10)).

Euler transform of length 6 sequence [1, -1, 1, 0, 1, -1].

G.f.: (1 + x) * (1 + x^3) / (1 - x^5).

a(n) = a(-n) = a(n + 5) = A011558(n + 3) for all n in Z.

Period 5 sequence [1, 1, 0, 1, 1, ...].

a(n) = A130782(n) mod 2. - Antti Karttunen, Aug 31 2017