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 1-6 of 6 results.

A291681 First differences of A067046.

Original entry on oeis.org

1, 8, 0, 25, -7, 56, -24, 105, -55, 176, -104, 273, -175, 400, -272, 561, -399, 760, -560, 1001, -759, 1288, -1000, 1625, -1287, 2016, -1624, 2465, -2015, 2976, -2464, 3553, -2975, 4200, -3552, 4921, -4199, 5720, -4920, 6601, -5719, 7568, -6600, 8625, -7567, 9776, -8624
Offset: 1

Views

Author

Enrique Navarrete, Sep 04 2017

Keywords

Comments

a(2n) > 0 and a(2n+1) < 0 for all n >= 2.

Links

Crossrefs

Cf. A067046.

Programs

  • Maple
    a:= n-> `if`(irem(n-1, 2, 'r')=0, -(r-1)*(2*r+3)*(r+1)/3
                                , (2*r+3)*(r+4)*(r+2)/3):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 04 2017
  • PARI
    Vec(x*(1 + 9*x + 5*x^2 - 2*x^3 - 3*x^4 + x^5 + x^6) / ((1 - x)^3*(1 + x)^4) + O(x^60)) \\ Colin Barker, Sep 29 2017

Formula

a(n) = A067046(n+1) - A067046(n).
G.f.: -x*(x^6+x^5-3*x^4-2*x^3+5*x^2+9*x+1)/((x-1)^3*(x+1)^4). - Alois P. Heinz, Sep 04 2017
From Colin Barker, Sep 29 2017: (Start)
a(n) = (n^3 + 9*n^2 + 20*n + 12) / 12 for n even.
a(n) = (-n^3 + 7*n + 6) / 12 for n odd.
(End)

A033931 a(n) = lcm(n,n+1,n+2).

Original entry on oeis.org

6, 12, 60, 60, 210, 168, 504, 360, 990, 660, 1716, 1092, 2730, 1680, 4080, 2448, 5814, 3420, 7980, 4620, 10626, 6072, 13800, 7800, 17550, 9828, 21924, 12180, 26970, 14880, 32736, 17952, 39270, 21420, 46620, 25308, 54834, 29640, 63960, 34440, 74046, 39732, 85140
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Also denominator of h(n+2) - h(n-1), where h(n) is the n-th harmonic number Sum_{k=1..n} 1/k, the numerator is A188386. - Reinhard Zumkeller, Jul 04 2012

Links

Crossrefs

Cf. A001008, A002805, A067046, A078637, A007947, A188386, A244009.

Programs

  • Haskell
    a033931 n = lcm n (lcm (n + 1) (n + 2))  -- Reinhard Zumkeller, Jul 04 2012
  • Magma
    [Numerator((n^3-n)/(n^2+1)): n in [2..50]]; // Vincenzo Librandi, Aug 19 2014
  • Maple
    a:= n-> ilcm($n..n+2):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 18 2025
  • Mathematica
    LCM@@@Partition[Range[50],3,1] (* or *) LinearRecurrence[{0,4,0,-6,0,4,0,-1},{6,12,60,60,210,168,504,360},50] (* Harvey P. Dale, Jun 29 2019 *)
  • PARI
    a(n) = lcm(n^2+n,n+2) \\ Charles R Greathouse IV, Sep 30 2016

Formula

a(n) = n*(n+1)*(n+2)*[3-(-1)^n]/4.
From Reinhard Zumkeller, Jul 04 2012: (Start)
a(n) = 6 * A067046(n).
A007947(a(n)) = A078637(n). (End)
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 - log(2) (A244009).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) - 2. (End)

A067047 a(n) = lcm(n, n+1, n+2, n+3)/12.

Original entry on oeis.org

1, 5, 5, 35, 70, 42, 210, 330, 165, 715, 1001, 455, 1820, 2380, 1020, 3876, 4845, 1995, 7315, 8855, 3542, 12650, 14950, 5850, 20475, 23751, 9135, 31465, 35960, 13640, 46376, 52360, 19635, 66045, 73815, 27417, 91390, 101270, 37310, 123410, 135751, 49665, 163185
Offset: 1

Views

Author

Amarnath Murthy, Dec 30 2001

Keywords

Examples

			a(6) = 42 as lcm(6,7,8,9)/12 = 72*7/12 = 42.

Links

Crossrefs

Cf. A067046.

Programs

  • Mathematica
    Table[LCM@@Range[n,n+3]/12,{n,40}] (* or *) LinearRecurrence[{0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1},{1,5,5,35,70,42,210,330,165,715,1001,455,1820,2380,1020},40] (* Harvey P. Dale, Dec 04 2016 *)
  • PARI
    a(n) = {lcm([n,n+1,n+2,n+3])/12} \\ Harry J. Smith, May 01 2010
  • PARI
    a(n)=binomial(n+3,4)/if(n%3,1,3) \\ Charles R Greathouse IV, Feb 28 2012

Formula

Quasipolynomial: a(n) = n(n+1)(n+2)(n+3)/72 if 3|n and a(n) = n(n+1)(n+2)(n+3)/24 otherwise.
a(n) = n*(n+1)*(n+2)*(n+3)/(8*(5+4*cos(2*n*Pi/3))). - Gary Detlefs, Apr 01 2011
G.f.: -x*(x^10 + 5*x^9 + 5*x^8 + 30*x^7 + 45*x^6 + 17*x^5 + 45*x^4 + 30*x^3 + 5*x^2 + 5*x+1)/ ((x-1)^5*(x^2+x+1)^5). - Colin Barker, Jul 01 2012
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 16 - 8*Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 160*log(2)/3 - 36. (End)

A067049 Triangle T(n,r) = lcm(n,n-1,n-2,...,n-r+1)/lcm(1,2,3,...,r-1,r), 0 <= r < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 2, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 10, 5, 1, 1, 1, 7, 21, 35, 35, 7, 7, 1, 1, 8, 28, 28, 70, 14, 14, 2, 1, 1, 9, 36, 84, 42, 42, 42, 6, 3, 1, 1, 10, 45, 60, 210, 42, 42, 6, 3, 1, 1, 1, 11, 55, 165, 330, 462, 462, 66, 33, 11, 11, 1, 1, 12, 66, 110
Offset: 0

Views

Author

Amarnath Murthy, Dec 30 2001

Keywords

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 4, 6, 2, 1; ...

References

  • Amarnath Murthy, Some Notions on Least Common Multiples, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Crossrefs

Diagonals give A067046, A067047, A067048. Row sums give A061297.

Programs

Extensions

More terms from Vladeta Jovovic, Dec 31 2001

A067048 a(n) = lcm(n, n+1, n+2, n+3, n+4) / 60.

Original entry on oeis.org

1, 1, 7, 14, 42, 42, 462, 66, 429, 1001, 1001, 364, 6188, 1428, 3876, 3876, 6783, 4389, 33649, 3542, 17710, 32890, 26910, 8190, 118755, 23751, 56637, 50344, 79112, 46376, 324632, 31416, 145299, 250971, 191919, 54834, 749398, 141778, 320866, 271502, 407253
Offset: 1

Views

Author

Amarnath Murthy, Dec 30 2001

Keywords

Examples

			a(6) = 42 as lcm(6,7,8,9,10)/60 = 2520/60 = 42.

Links

Crossrefs

Cf. A067046, A067047.

Programs

  • Maple
    seq(ilcm(n,n+1,n+2,n+3,n+4)/60,n=1..100); # Robert Israel, Feb 07 2016
  • Mathematica
    Table[LCM @@ Range[n, n + 4]/60, {n, 1, 50}] (* Amiram Eldar, Sep 29 2022 *)
  • PARI
    a(n)={lcm([n, n+1, n+2, n+3, n+4])/60} \\ Harry J. Smith, May 01 2010

Formula

From Gary Detlefs Apr 14 2011 and Apr 18 2011: (Start)
a(n) = (n+4)!*gcd(n-1,3)/(360*(n-1)!*gcd(n,4))
a(n) = (n+4)!*(5-4*cos((2*n+1)*Pi/3))/(1080*(n-1)!*(2+(-1)^n+cos(n*Pi/2)))
a(n) = (n+4)!*gcd(n-1,6)/(180*(n-1)!*2^((2*cos(n*Pi/2)+9+(-1)^n)/4)), n>1. (End)
120 <= n*(n+1)*(n+2)*(n+3)*(n+4)/a(n) <= 1440. - Charles R Greathouse IV, Sep 19 2012
Sum_{n>=1} 1/a(n) = 80 - 40*log(sqrt(3)+2)/sqrt(3) - 490*log(2)/3 + 60*log(3). - Amiram Eldar, Sep 29 2022

A189046 a(n) = lcm(n,n+1,n+2,n+3,n+4,n+5)/60.

Original entry on oeis.org

0, 1, 7, 14, 42, 42, 462, 462, 858, 3003, 1001, 4004, 6188, 18564, 27132, 3876, 27132, 74613, 100947, 67298, 17710, 230230, 296010, 188370, 237510, 118755, 736281, 453096, 553784, 1344904, 324632
Offset: 0

Views

Author

Gary Detlefs, Apr 15 2011

Keywords

Comments

a(n) mod 2 has a period of 8, repeating [0,1,1,0,0,0,0,0].

Links

Crossrefs

Cf. A000217 ( = lcm(n,n+1)/2), A021913, A067046, A067047, A067048.

Programs

  • Maple
    seq(lcm(n,n+1,n+2,n+3,n+4,n+5)/60,n=0..30)
  • Mathematica
    Table[(LCM@@(n+Range[0,5]))/60,{n,0,40}]  (* Harvey P. Dale, Apr 17 2011 *)
  • PARI
    a(n)=lcm([n..n+5])/60 \\ Charles R Greathouse IV, Sep 30 2016

Formula

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(4*(n^4 mod 5)+1)/(1800*((n^3 mod 4)+((n-1)^3 mod 4)+1)).
a(n) = binomial(n+5,6)/(gcd(n,5)*(A021913(n-1)+1)).
a(n) = binomial(n+5,6)/(gcd(n,5)*floor(((n-1) mod 4)/2+1)). - Gary Detlefs, Apr 22 2011
Sum_{n>=1} 1/a(n) = 92 + (54/5-18*sqrt(5)+6*sqrt(178-398/sqrt(5)))*Pi. - Amiram Eldar, Sep 29 2022
Showing 1-6 of 6 results.

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