A047238 - OEIS

%I #79 Jul 12 2022 20:59:04

%S 0,2,6,8,12,14,18,20,24,26,30,32,36,38,42,44,48,50,54,56,60,62,66,68,

%T 72,74,78,80,84,86,90,92,96,98,102,104,108,110,114,116,120,122,126,

%U 128,132,134,138,140,144,146,150,152,156,158,162

%N Numbers that are congruent to {0, 2} mod 6.

%C Complement of A047251, or "Polyrhythmic Sequence" P(2,3); the present sequence represents where the "rests" occur in a "3 against 2" polyrhythm. (See A267027 for definition and description). - _Bob Selcoe_, Jan 12 2016

%H Bruno Berselli, <a href="/A047238/b047238.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F From _Bruno Berselli_, Jun 24 2010: (Start)

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

%F a(n) = a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=2, a(2)=6.

%F a(n) = (6*n - (-1)^n-7)/2.

%F a(n) = 2*A032766(n-1). (End)

%F a(n) = 6*n - a(n-1) - 10 (with a(1)=0). - _Vincenzo Librandi_, Aug 05 2010

%F a(n+1) = Sum_{k>=0} A030308(n,k)*A111286(k+2). - _Philippe Deléham_, Oct 17 2011

%F a(n) = 2*floor(3*n/2). - _Enrique Pérez Herrero_, Jul 04 2012

%F Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4. - _Amiram Eldar_, Dec 13 2021

%F E.g.f: 3*(x-1)*exp(x) - cosh(x) + 4. - _David Lovler_, Jul 11 2022

%t Select[Range[0,200],MemberQ[{0,2},Mod[#,6]]&] (* or *) LinearRecurrence[ {1,1,-1},{0,2,6},70] (* _Harvey P. Dale_, Jun 15 2011 *)

%o (PARI) forstep(n=0,200,[2,4],print1(n", ")) \\ _Charles R Greathouse IV_, Oct 17 2011

%o (Magma) [n: n in [0..200]|n mod 6 in {0,2}]; // _Vincenzo Librandi_, Jan 12 2016

%Y Cf. A047270 [(6*n-(-1)^n-1)/2], A047235 [(6*n-(-1)^n-3)/2], A047241 [(6*n-(-1)^n-5)/2].

%Y Cf. A047251, A267027.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_