A047263 Numbers that are congruent to {0, 1, 2, 4, 5} mod 6.

0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

Complement of A016945. - R. J. Mathar, Feb 25 2008

Nonnegative integers m such that floor(2*m^2/12) = 2*floor(m^2/12). See the Crossrefs field of A265187 for similar sequences. - Bruno Berselli, Dec 08 2015

Also, numbers k such that Fibonacci(k) mod 4 = 0, 1 or 3. - Bruno Berselli, Oct 17 2017

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Cf. A016945, A265187.

Programs

Magma

[n : n in [0..100] | n mod 6 in [0, 1, 2, 4, 5]]; // Wesley Ivan Hurt, Aug 16 2016

Maple

for n from 0 to 200 do if n mod 6 <> 3 then printf(`%d,`,n) fi: od:
A047263:=n->6*floor(n/5)+[0, 1, 2, 4, 5][(n mod 5)+1]: seq(A047263(n), n=0..100); # Wesley Ivan Hurt, Aug 16 2016

Mathematica

Select[Range[0,100], Mod[#,6]!=3&] (* Harvey P. Dale, May 17 2011 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,4,5,6},90] (* Harvey P. Dale, Oct 05 2014 *)

PARI

first(n) = {select(x->(x%6!=3), vector(6*n\5, i, i-1))} \\ David A. Corneth, Oct 17 2017

Formula

O.g.f.: x*(x^2+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - R. J. Mathar, Feb 25 2008

a(n) = a(n-5) + 6 for n > 5. - R. J. Mathar, Feb 25 2008

a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6. - Harvey P. Dale, Oct 05 2014

From Wesley Ivan Hurt, Aug 16 2016: (Start)

a(n) = n + floor((n-4)/5).

a(n) = (6*n - 4 - ((n+1) mod 5))/5.

a(5k) = 6k-1, a(5k-1) = 6k-2, a(5k-2) = 6k-4, a(5k-3) = 6k-5, a(5k-4) = 6k-6. (End)

Sum_{n>=2} (-1)^n/a(n) = log(2+sqrt(3))/sqrt(3) - log(2)/6. - Amiram Eldar, Dec 17 2021

Extensions

More terms from James Sellers, Feb 19 2001