A281746 Nonnegative numbers k such that k == 0 (mod 3) or k == 0 (mod 5).

0, 3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 125, 126, 129, 130, 132, 135

Offset: 1

Author

Seiichi Manyama, Jan 29 2017

Keywords

• nonn
• easy

Comments

In the game "FizzBuzz", players replace any number divisible by three with the word "Fizz", and any number divisible by five with the word "Buzz". But multiples of both three and five are replaced by "FizzBuzz". For example, 1, 2, Fizz, 4, Buzz, Fizz, 7, 8, Fizz, Buzz, 11, Fizz, 13, 14, FizzBuzz, 16, 17, Fizz, 19, Buzz, Fizz, 22, 23, Fizz, Buzz, 26, Fizz, 28, 29, FizzBuzz, ...

The asymptotic density of this sequence is 7/15. - Amiram Eldar, Mar 25 2021

For a neat way to supplement the set to achieve equal density with its complement, see A080307. - Peter Munn, Oct 12 2023

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Project Euler, Problem 1: Multiples of 3 or 5.
• Rosetta Code, FizzBuzz.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Complement of A229829.

Union of A008585 and A008587.

Subsequence of {0} U A080307.

Cf. A281787.

Programs

Mathematica

Select[Range[0, 200], MemberQ[Mod[#, {3, 5}], 0]&] (* or *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 3, 5, 6, 9, 10, 12, 15}, 80] (* Harvey P. Dale, Apr 01 2018 *)
Union[3Range[0, 33], 5Range[20]] (* Alonso del Arte, Sep 03 2018 *)
CoefficientList[Series[-(3*x^7 + 2*x^6 + x^5 + 3*x^4 + x^3 + 2*x^2 + 3*x) / (-x^8 + x^7 + x - 1) , {x, 0, 80}], x] (* Stefano Spezia, Sep 16 2018 *)

PARI

concat(0, Vec(x^2*(3 + 2*x + x^2 + 3*x^3 + x^4 + 2*x^5 + 3*x^6) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^100))) \\ Colin Barker, Feb 07 2017

Formula

G.f.: -(3*x^8 + 2*x^7 + x^6 + 3*x^5 + x^4 + 2*x^3 + 3*x^2) / (-x^8 + x^7 + x - 1).

a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8. - Colin Barker, Feb 07 2017

a(n) = 15n/7 + O(1). - Charles R Greathouse IV, Jan 13 2025