A047273 Numbers that are congruent to {0, 1, 3, 5} mod 6.

0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A047235. - Reinhard Zumkeller, Oct 01 2008

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Index entries for two-way infinite sequences.
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A004524, A007310, A047228, A047235, A047241, A047261, A093719, A096981.

First differences of A281026.

See A301729 for an essentially identical sequence.

Programs

Haskell

a047273 n = a047273_list !! (n-1)
a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list
-- Reinhard Zumkeller, Feb 19 2013

Magma

[(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 20 2016

Maple

seq(2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, n = 0..69); # Gary Detlefs, Mar 19 2010

Mathematica

LinearRecurrence[{2,-2,2,-1},{0,1,3,5},80] (* Harvey P. Dale, Jan 04 2015 *)

PARI

a(n)=n+(n+1)\4+(n+2)\4

Sage

[(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # Zerinvary Lajos, Mar 09 2009

Formula

G.f.: x*(1+x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^2)*(1-x^3)/((1-x)^3*(1-x^4)).

a(n) = n + A004524(n+1) = -a(-n) for all n in Z.

Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - Gary W. Adamson, Jun 19 2008

A093719(a(n)) = 1. - Reinhard Zumkeller, Oct 01 2008

a(n) = 2*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/4, with offset 0..a(0)=0. - Gary Detlefs, Mar 19 2010

a(n) = (3*n-3+cos(Pi*n/2))/2. - R. J. Mathar, Oct 08 2010

From Wesley Ivan Hurt, May 20 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-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)

Euler transform of length 4 sequence [3, -1, -1, 1]. - Michael Somos, Jun 24 2017

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

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