A249860 a(n) = Least common multiple of n + 3 and n - 3.

4, 5, 0, 7, 8, 9, 20, 55, 12, 91, 56, 45, 80, 187, 36, 247, 140, 105, 176, 391, 72, 475, 260, 189, 308, 667, 120, 775, 416, 297, 476, 1015, 180, 1147, 608, 429, 680, 1435, 252, 1591, 836, 585, 920, 1927, 336, 2107, 1100, 765, 1196, 2491, 432, 2695, 1400, 969

Offset: 1

Author

Colin Barker, Nov 07 2014

Keywords

• nonn
• easy

Comments

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.

Examples

			a(8) = 55 because lcm(8+3, 8-3) = lcm(11, 5) = 55.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Crossrefs

Cf. A066830 (k=1), A249859 (k=2), A060789.

Programs

Maple

A249860:=n->lcm(n+3,n-3): seq(A249860(n), n=1..100); # Wesley Ivan Hurt, Feb 12 2017

Mathematica

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 *)
Table[LCM @@ (n + {-3, 3}), {n, 54}] (* Michael De Vlieger, Feb 13 2017 *)

PARI

a(n) = lcm(n+3, n-3)

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))

Formula

a(n) = 3*a(n-6)-3*a(n-12)+a(n-18) for n>18.

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).

From Peter Bala, Feb 15 2019: (Start)

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

For n >= 4, a(n) = (n + 3)*A060789(n-3). (End)

Sum_{n>=4} 1/a(n) = 47/60. - Amiram Eldar, Aug 09 2022

Sum_{k=1..n} a(k) ~ 7*n^3/36. - Vaclav Kotesovec, Aug 09 2022