A047261 Numbers that are congruent to {2, 4, 5} mod 6.

2, 4, 5, 8, 10, 11, 14, 16, 17, 20, 22, 23, 26, 28, 29, 32, 34, 35, 38, 40, 41, 44, 46, 47, 50, 52, 53, 56, 58, 59, 62, 64, 65, 68, 70, 71, 74, 76, 77, 80, 82, 83, 86, 88, 89, 92, 94, 95, 98, 100, 101, 104, 106, 107, 110, 112, 113, 116, 118, 119, 122, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If B and C are terms in the sequence then 2*B*C is a term. B (resp. C) is a term iff B (resp. C) mod 6 = 2, 4 or 5. It follows that (2*B*C) mod 6 = (2*(B mod 6)*(C mod 6)) mod 6 = 2 or 4 and therefore 2*B*C is a term. Examples: for B=16 and C=29, 2*16*29 = 928 is a term: (2*B*C) mod 6 = (2*16*29) mod 6 = 4; (2*2*2) mod 6 = 2. - Jerzy R Borysowicz, May 24 2018

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A047242 (complement).

Cf. A007310, A047228, A047241, A047273, A056970, A214090.

Programs

Haskell

a047261 n = a047261_list !! n
a047261_list = 2 : 4 : 5 : map (+ 6) a047261_list
-- Reinhard Zumkeller, Feb 19 2013, Jul 06 2012

Magma

[n : n in [0..150] | n mod 6 in [2, 4, 5]]; // Wesley Ivan Hurt, Jun 14 2016

Maple

A047261:=n->(6*n-1-2*cos(2*n*Pi/3))/3: seq(A047261(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Mathematica

CoefficientList[Series[(1 + x)*(x^2 + 2)/((1 + x + x^2)*(x - 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Aug 16 2014 *)
Select[ Range@ 125, MemberQ[{2, 4, 5}, Mod[#, 6]] &] (* or *)
LinearRecurrence[{1, 0, 1, -1}, {2, 4, 5, 8}, 62] (* Robert G. Wilson v, Jun 13 2018 *)

Formula

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

A214090(a(n)) = 1. - Reinhard Zumkeller, Jul 06 2012

From Wesley Ivan Hurt, Jun 14 2016: (Start)

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

a(n) = (6*n - 1 - 2*cos(2*n*Pi/3))/3.

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

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2+sqrt(3))/(2*sqrt(3)) + log(2)/3. - Amiram Eldar, Dec 16 2021

E.g.f.: (3 + exp(x)*(6*x - 1) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 26 2024

Extensions

More terms from Wesley Ivan Hurt, Aug 16 2014