A047273 - OEIS

%I #56 Jan 04 2025 02:41:18

%S 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,

%T 37,39,41,42,43,45,47,48,49,51,53,54,55,57,59,60,61,63,65,66,67,69,71,

%U 72,73,75,77,78,79,81,83,84,85,87,89,90,91,93,95,96,97,99,101,102,103

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

%C Complement of A047235. - _Reinhard Zumkeller_, Oct 01 2008

%H Reinhard Zumkeller, <a href="/A047273/b047273.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F 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)).

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

%F Starting (1, 3, 5, ...) = partial sums of (1, 2, 2, 1, 1, 2, 2, 1, 1, ...). - _Gary W. Adamson_, Jun 19 2008

%F A093719(a(n)) = 1. - _Reinhard Zumkeller_, Oct 01 2008

%F 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

%F a(n) = (3*n-3+cos(Pi*n/2))/2. - _R. J. Mathar_, Oct 08 2010

%F From _Wesley Ivan Hurt_, May 20 2016: (Start)

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

%F a(n) = (6*n-6+(-1)^(n/2)+(-1)^(-n/2))/4. (End)

%F Euler transform of length 4 sequence [3, -1, -1, 1]. - _Michael Somos_, Jun 24 2017

%F Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(3)/2. - _Amiram Eldar_, Dec 16 2021

%F E.g.f.: (2 + 3*exp(x)*(x - 1) + cos(x))/2. - _Stefano Spezia_, Jul 26 2024

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

%t LinearRecurrence[{2,-2,2,-1},{0,1,3,5},80] (* _Harvey P. Dale_, Jan 04 2015 *)

%o (PARI) a(n)=n+(n+1)\4+(n+2)\4

%o (Sage) [(lucas_number1(n+2, 0, 1)+3*n)/2 for n in range(0, 70)] # _Zerinvary Lajos_, Mar 09 2009

%o (Haskell)

%o a047273 n = a047273_list !! (n-1)

%o a047273_list = 0 : 1 : 3 : 5 : map (+ 6) a047273_list

%o -- _Reinhard Zumkeller_, Feb 19 2013

%o (Magma) [(6*n-6+(-1)^(n div 2)+(-1)^(-n div 2))/4: n in [1..100]]; // _Wesley Ivan Hurt_, May 20 2016

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

%Y First differences of A281026.

%Y See A301729 for an essentially identical sequence.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_