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%I #30 Aug 10 2022 07:55:52
%S 4,5,0,7,8,9,20,55,12,91,56,45,80,187,36,247,140,105,176,391,72,475,
%T 260,189,308,667,120,775,416,297,476,1015,180,1147,608,429,680,1435,
%U 252,1591,836,585,920,1927,336,2107,1100,765,1196,2491,432,2695,1400,969
%N a(n) = Least common multiple of n + 3 and n - 3.
%C The recurrence for the general case lcm(n+k, n-k) is a(n) = 3*a(n-2*k)-3*a(n-4*k)+a(n-6*k) for n>6*k.
%H Colin Barker, <a href="/A249860/b249860.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).
%F a(n) = 3*a(n-6)-3*a(n-12)+a(n-18) for n>18.
%F G.f.: x*(-10*x^19 -8*x^18 -3*x^17 -4*x^16 -5*x^15 +37*x^13 +32*x^12 +18*x^11 +32*x^10 +70*x^9 +12*x^8 +40*x^7 +8*x^6 +9*x^5 +8*x^4 +7*x^3 +5*x +4) / (-x^18 +3*x^12 -3*x^6 +1).
%F From _Peter Bala_, Feb 15 2019: (Start)
%F For n >= 3, a(n) = (n^2 - 9)/b(n), where (b(n)), n >= 3, is the periodic sequence [6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, ...] of period 6. a(n) is thus a quasi-polynomial in n
%F For n >= 4, a(n) = (n + 3)*A060789(n-3). (End)
%F Sum_{n>=4} 1/a(n) = 47/60. - _Amiram Eldar_, Aug 09 2022
%F Sum_{k=1..n} a(k) ~ 7*n^3/36. - _Vaclav Kotesovec_, Aug 09 2022
%e a(8) = 55 because lcm(8+3, 8-3) = lcm(11, 5) = 55.
%p A249860:=n->lcm(n+3,n-3): seq(A249860(n), n=1..100); # _Wesley Ivan Hurt_, Feb 12 2017
%t CoefficientList[Series[(-10 x^19 - 8 x^18 - 3 x^17 - 4 x^16 -5 x^15 + 37 x^13 + 32 x^12 + 18 x^11 + 32 x^10 + 70 x^9 + 12 x^8 + 40 x^7 + 8 x^6 + 9 x^5 + 8 x^4 + 7 x^3 + 5 x + 4) / (- x^18 + 3 x^12 - 3 x^6 + 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 08 2014 *)
%t Table[LCM @@ (n + {-3, 3}), {n, 54}] (* _Michael De Vlieger_, Feb 13 2017 *)
%o (PARI) a(n) = lcm(n+3, n-3)
%o (PARI) Vec(x*(-10*x^19 -8*x^18 -3*x^17 -4*x^16 -5*x^15 +37*x^13 +32*x^12 +18*x^11 +32*x^10 +70*x^9 +12*x^8 +40*x^7 +8*x^6 +9*x^5 +8*x^4 +7*x^3 +5*x +4) / (-x^18 +3*x^12 -3*x^6 +1) + O(x^100))
%Y Cf. A066830 (k=1), A249859 (k=2), A060789.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Nov 07 2014