A047262 Numbers that are congruent to {0, 2, 4, 5} mod 6.

0, 2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 66, 68, 70, 71, 72, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence is the interleaving of A047233 with A016789(n-1). - Guenther Schrack, Feb 14 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
• Index entries for two-way infinite sequences.

Crossrefs

Cf. A016789, A047233, A047237, A047273.

Complement: A047241.

Programs

Magma

[n : n in [0..100] | n mod 6 in [0, 2, 4, 5]]; // Wesley Ivan Hurt, May 21 2016

Maple

A047262:=n->(6*n-4-I^(1-n)+I^(1+n))/4: seq(A047262(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Mathematica

Select[Range[0,100], MemberQ[{0,2,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{2,-2,2,-1}, {0,2,4,5}, 70] (* Harvey P. Dale, Dec 09 2015 *)

PARI

my(x='x+O('x^70)); concat([0], Vec(x^2*(2+x^2)/((1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019

Sage

a=(x^2*(2+x^2)/((1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Formula

From R. J. Mathar, Oct 08 2011: (Start)

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

a(n) = 3*n/2 - 1 - sin(Pi*n/2)/2. (End)

From Wesley Ivan Hurt, May 21 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4.

a(n) = (6*n - 4 - i^(1-n) + i^(1+n))/4, where i = sqrt(-1).

a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047233(n).

a(2-n) = - A047237(n), a(n-1) = A047273(n) - 1 for n > 1. (End)

From Guenther Schrack, Feb 14 2019: (Start)

a(n) = (6*n - 4 - (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.

a(n) = a(n-4) + 6, a(1)=0, a(2)=2, a(3)=4, a(4)=5, for n > 4. (End)

Sum_{n>=2} (-1)^n/a(n) = log(3)/4 + log(2)/3 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 17 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016