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a(n) = lcm(n, n+1, n+2)/6.
7

%I #49 Dec 04 2024 16:49:32

%S 1,2,10,10,35,28,84,60,165,110,286,182,455,280,680,408,969,570,1330,

%T 770,1771,1012,2300,1300,2925,1638,3654,2030,4495,2480,5456,2992,6545,

%U 3570,7770,4218,9139,4940,10660,5740,12341,6622,14190,7590,16215,8648,18424,9800

%N a(n) = lcm(n, n+1, n+2)/6.

%H Harry J. Smith, <a href="/A067046/b067046.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).

%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.

%F G.f.: (x^4 + 2x^3 + 6x^2 + 2x + 1)/(1 - x^2)^4.

%F a(n) = binomial(n+2,3)*(3-(-1)^n)/4. - _Gary Detlefs_, Apr 13 2011

%F Quasipolynomial: a(n) = n(n+1)(n+2)/6 when n is odd and n(n+1)(n+2)/12 otherwise. - _Charles R Greathouse IV_, Feb 27 2012

%F a(n) = A033931(n) / 6. - _Reinhard Zumkeller_, Jul 04 2012

%F From _Amiram Eldar_, Sep 29 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 6*(1 - log(2)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(3*log(2) - 2). (End)

%e a(6) = 28 as lcm(6,7,8)/6 = 168/6 = 28.

%t Table[LCM[n,n+1,n+2]/6,{n,50}] (* _Harvey P. Dale_, Jan 11 2011 *)

%o (PARI) a(n)={lcm([n, n+1, n+2])/6} \\ _Harry J. Smith_, Apr 30 2010

%o (PARI) a(n)=binomial(n+2,3)/(2-n%2) \\ _Charles R Greathouse IV_, Feb 27 2012

%o (Haskell)

%o a067046 = (`div` 6) . a033931 -- _Reinhard Zumkeller_, Jul 04 2012

%Y Cf. A000447 (bisection), A006331 (bisection), A033931.

%K nonn,easy

%O 1,2

%A _Amarnath Murthy_, Dec 30 2001