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 1-4 of 4 results.

A276705 Records in A249860.

Original entry on oeis.org

4, 5, 7, 8, 9, 20, 55, 91, 187, 247, 391, 475, 667, 775, 1015, 1147, 1435, 1591, 1927, 2107, 2491, 2695, 3127, 3355, 3835, 4087, 4615, 4891, 5467, 5767, 6391, 6715, 7387, 7735, 8455, 8827, 9595, 9991, 10807, 11227, 12091, 12535, 13447, 13915, 14875, 15367
Offset: 1

Views

Author

Colin Barker, Sep 15 2016

Keywords

Comments

The indices corresponding to the records are in A276706.

Crossrefs

Cf. A249860, A276704, A276706.

Programs

  • Mathematica
    DeleteDuplicates[Table[LCM@@(n+{-3,3}),{n,200}],GreaterEqual] (* Harvey P. Dale, Jul 30 2025 *)
  • PARI
    lista(nn) = {rec = 0; for (n=1, nn, an = lcm(n-3, n+3); if (an > rec, rec = an; print1(rec, ", ")););} \\ Michel Marcus, Sep 16 2016

Formula

Conjectures: (Start)
a(n) = (347+27*(-1)^n)/2 - 3*(27+(-1)^n)*n + 9*n^2 for n>6.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>11.
G.f.: x*(4+x-6*x^2-x^3+x^4+10*x^5+35*x^6+15*x^7+27*x^8-x^9-13*x^10) / ((1-x)^3*(1+x)^2).
(End)

A276706 Indices of records in A249860.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166
Offset: 1

Views

Author

Colin Barker, Sep 15 2016

Keywords

Comments

The records corresponding to the indices are in A276705.

Crossrefs

Cf. A249860, A276705.

Programs

  • PARI
    lista(nn) = {rec = 0; for (n=1, nn, an = lcm(n-3, n+3); if (an > rec, rec = an; print1(n, ", ")););} \\ Michel Marcus, Sep 16 2016

Formula

Conjectures: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
a(n) = (6*n-27-(-1)^n)/2 for n>6.
G.f.: x*(1+x+x^2-x^4+x^7+3*x^8) / ((1-x)^2*(1+x)).
(End)

A249859 Least common multiple of n + 2 and n - 2.

Original entry on oeis.org

3, 0, 5, 6, 21, 8, 45, 30, 77, 24, 117, 70, 165, 48, 221, 126, 285, 80, 357, 198, 437, 120, 525, 286, 621, 168, 725, 390, 837, 224, 957, 510, 1085, 288, 1221, 646, 1365, 360, 1517, 798, 1677, 440, 1845, 966, 2021, 528, 2205, 1150, 2397, 624, 2597, 1350, 2805
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) = 30 because lcm(8 + 2, 8 - 2) = lcm(6, 10) = 30.

Links

Crossrefs

Cf. A066830 (k=1), A249860 (k=3), A060819.

Programs

  • Magma
    [Lcm(n-2, n+2): n in [1..60]]; // Vincenzo Librandi, Nov 10 2014
  • Maple
    A249859:=n-> ilcm(n+2,n-2): seq(A249859(n), n=1..100); # Wesley Ivan Hurt, Jul 09 2017
  • Mathematica
    Table[LCM[n - 2, n + 2], {n, 50}] (* Alonso del Arte, Nov 07 2014 *)
CoefficientList[Series[(-6 x^12 - 2 x^11 - 3 x^10 + 23 x^8 + 12 x^7 + 30 x^6 + 8 x^5 + 12 x^4 + 6 x^3 + 5 x^2 + 3) / (-x^12 + 3 x^8 - 3 x^4 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{3,0,5,6,21,8,45,30,77,24,117,70,165},60] (* Harvey P. Dale, Jul 11 2017 *)
  • PARI
    a(n) = lcm(n+2, n-2)
  • PARI
    Vec(x*(-6*x^12 -2*x^11 -3*x^10 +23*x^8 +12*x^7 +30*x^6 +8*x^5 +12*x^4 +6*x^3 +5*x^2 +3) / (-x^12 +3*x^8 -3*x^4 +1) + O(x^100))

Formula

a(n) = lcm(n - 2, n + 2).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n > 12.
G.f.: x*(-6*x^12 - 2*x^11 - 3*x^10 + 23*x^8 + 12*x^7 + 30*x^6 + 8*x^5 + 12*x^4 + 6*x^3 + 5*x^2 + 3) / (-x^12 + 3*x^8 - 3*x^4 + 1).
From Peter Bala, Feb 15 2019: (Start)
For n >= 2, a(n) = (n^2 - 4)/b(n), where b(n), n >= 1, is the periodic sequence [1, 4, 1, 2, 1, 4, 1, 2, ...] of period 4. a(n) is thus a quasi-polynomial in n.
For n >= 3, a(n) = (n + 2)*A060819(n-2). (End)
Sum_{n>=3} 1/a(n) = 5/6. - Amiram Eldar, Aug 09 2022
Sum_{k=1..n} a(k) ~ 11*n^3/48. - 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 1-4 of 4 results.

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