A047237 Numbers that are congruent to {0, 1, 2, 4} mod 6.

0, 1, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 97

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence is the interleaving of A047238(n) with A016777(n-1). - Guenther Schrack, Feb 11 2019

Links

• Guenther Schrack, Table of n, a(n) for n = 1..10002
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A016777, A047238, A047255, A047262, A118286.

Complement: A047270.

Programs

GAP

Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 4); # Muniru A Asiru, Feb 19 2019

Magma

[n : n in [0..110] | n mod 6 in [0, 1, 2, 4]]; // G. C. Greubel, Feb 16 2019

Maple

A047237:=n->(6*n-8+I^(1-n)-I^(1+n))/4: seq(A047237(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Mathematica

Table[(6n-8+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{2,-2,2,-1},{0,1,2,4},120] (* Harvey P. Dale, Jan 21 2018 *)

PARI

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

Sage

a=(x^2*(1+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

Starting (1, 2, 4, 6, ...) = partial sums of (1, 1, 2, 2, 1, 1, 2, 2, ...). - Gary W. Adamson, Jun 19 2008

G.f.: x^2*(1+2*x^2) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011

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 - 8 + i^(1-n) - i^(1+n))/4 where i=sqrt(-1).

a(2*n) = A016777(n-1), a(2*n-1) = A047238(n). (End)

From Guenther Schrack, Feb 11 2019: (Start)

a(n) = (6*n - 8 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.

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

a(-n) = -A047262(n+2).

a(n) = A118286(n-1)/2 for n > 1.

a(n) = A047255(n) - 1. (End)

Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/3 + log(3)/4. - Amiram Eldar, Dec 16 2021

Extensions

More terms from Wesley Ivan Hurt, May 21 2016