cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A276704 Records in A249859.

Original entry on oeis.org

3, 5, 6, 21, 45, 77, 117, 165, 221, 285, 357, 437, 525, 621, 725, 837, 957, 1085, 1221, 1365, 1517, 1677, 1845, 2021, 2205, 2397, 2597, 2805, 3021, 3245, 3477, 3717, 3965, 4221, 4485, 4757, 5037, 5325, 5621, 5925, 6237, 6557, 6885, 7221, 7565, 7917, 8277
Offset: 1

Views

Author

Colin Barker, Sep 15 2016

Keywords

Comments

The indices of the records seem to be 4 and the odd integers.

Crossrefs

Cf. A249859, A276705.

Formula

Conjectures: (Start)
a(n) = 4*n^2-12*n+5 for n>3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>6.
G.f.: x*(3-4*x+15*x^3-5*x^4-x^5) / (1-x)^3.
(End)

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

Original entry on oeis.org

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

Views

Author

Colin Barker, Nov 07 2014

Keywords

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

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

A277384 Least common multiple of n + 4 and n - 4.

Original entry on oeis.org

15, 6, 7, 0, 9, 10, 33, 12, 65, 42, 105, 16, 153, 90, 209, 60, 273, 154, 345, 48, 425, 234, 513, 140, 609, 330, 713, 96, 825, 442, 945, 252, 1073, 570, 1209, 160, 1353, 714, 1505, 396, 1665, 874, 1833, 240, 2009, 1050, 2193, 572, 2385, 1242, 2585, 336, 2793
Offset: 1

Views

Author

Colin Barker, Oct 12 2016

Keywords

Comments

The recurrence for the general case lcm(n+k, n-k) is b(n) = 3*b(n-2*k) - 3*b(n-4*k) + b(n-6*k) for n>6*k.

Links

Crossrefs

Cf. A066830 (k=1), A249859 (k=2), A249860 (k=3).
Cf. A277385.

Programs

  • Maple
    A277384:=n->lcm(n+4,n-4): seq(A277384(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2017
  • Mathematica
    Table[LCM[n + 4, n - 4], {n, 1, 25}] (* G. C. Greubel, Oct 12 2016 *)
  • PARI
    a(n) = lcm(n+4, n-4)
  • PARI
    Vec(x*(15 +6*x +7*x^2 +9*x^4 +10*x^5 +33*x^6 +12*x^7 +20*x^8 +24*x^9 +84*x^10 +16*x^11 +126*x^12 +60*x^13 +110*x^14 +24*x^15 +123*x^16 +46*x^17 +51*x^18 -7*x^20 -6*x^21 -15*x^22 -4*x^23 -30*x^24 -12*x^25 -14*x^26) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3) + O(x^60))

Formula

a(n) = 3*a(n-8)-3*a(n-16)+a(n-24) for n>27.
G.f.: x*(15 +6*x +7*x^2 +9*x^4 +10*x^5 +33*x^6 +12*x^7 +20*x^8 +24*x^9 +84*x^10 +16*x^11 +126*x^12 +60*x^13 +110*x^14 +24*x^15 +123*x^16 +46*x^17 +51*x^18 -7*x^20 -6*x^21 -15*x^22 -4*x^23 -30*x^24 -12*x^25 -14*x^26) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^3*(1 +x^4)^3).
Showing 1-3 of 3 results.

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