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A067047 a(n) = lcm(n, n+1, n+2, n+3)/12.

%I A067047 #41 Dec 04 2024 17:50:52
%S A067047 1,5,5,35,70,42,210,330,165,715,1001,455,1820,2380,1020,3876,4845,
%T A067047 1995,7315,8855,3542,12650,14950,5850,20475,23751,9135,31465,35960,
%U A067047 13640,46376,52360,19635,66045,73815,27417,91390,101270,37310,123410,135751,49665,163185
%N A067047 a(n) = lcm(n, n+1, n+2, n+3)/12.
%H A067047 Harry J. Smith, <a href="/A067047/b067047.txt">Table of n, a(n) for n = 1..1000</a>
%H A067047 Amarnath Murthy, <a href="http://fs.unm.edu/SN/SomeNotionsLeast.pdf">Some Notions on Least Common Multiples</a>, Smarandache Notions Journal, Vol. 12, No. 1-2-3 (Spring 2001), pp. 307-308.
%H A067047 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1).
%H A067047 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.
%F A067047 Quasipolynomial: a(n) = n(n+1)(n+2)(n+3)/72 if 3|n and a(n) = n(n+1)(n+2)(n+3)/24 otherwise.
%F A067047 a(n) = n*(n+1)*(n+2)*(n+3)/(8*(5+4*cos(2*n*Pi/3))). - _Gary Detlefs_, Apr 01 2011
%F A067047 G.f.: -x*(x^10 + 5*x^9 + 5*x^8 + 30*x^7 + 45*x^6 + 17*x^5 + 45*x^4 + 30*x^3 + 5*x^2 + 5*x+1)/ ((x-1)^5*(x^2+x+1)^5). - _Colin Barker_, Jul 01 2012
%F A067047 From _Amiram Eldar_, Sep 29 2022: (Start)
%F A067047 Sum_{n>=1} 1/a(n) = 16 - 8*Pi/sqrt(3).
%F A067047 Sum_{n>=1} (-1)^(n+1)/a(n) = 160*log(2)/3 - 36. (End)
%e A067047 a(6) = 42 as lcm(6,7,8,9)/12 = 72*7/12 = 42.
%t A067047 Table[LCM@@Range[n,n+3]/12,{n,40}] (* or *) LinearRecurrence[{0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1},{1,5,5,35,70,42,210,330,165,715,1001,455,1820,2380,1020},40] (* _Harvey P. Dale_, Dec 04 2016 *)
%o A067047 (PARI) a(n) = {lcm([n,n+1,n+2,n+3])/12} \\ _Harry J. Smith_, May 01 2010
%o A067047 (PARI) a(n)=binomial(n+3,4)/if(n%3,1,3) \\ _Charles R Greathouse IV_, Feb 28 2012
%Y A067047 Cf. A067046.
%K A067047 nonn,easy
%O A067047 1,2
%A A067047 _Amarnath Murthy_, Dec 30 2001