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.

A120112 Row sums of number triangle A120111.

Original entry on oeis.org

1, -1, -2, -1, -4, 0, -6, -1, -2, 0, -10, 0, -12, 0, 0, -1, -16, 0, -18, 0, 0, 0, -22, 0, -4, 0, -2, 0, -28, 0, -30, -1, 0, 0, 0, 0, -36, 0, 0, 0, -40, 0, -42, 0, 0, 0, -46, 0, -6, 0, 0, 0, -52, 0, 0, 0, 0, 0, -58, 0, -60, 0, 0, -1, 0, 0, -66, 0, 0, 0, -70, 0, -72
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

The last row of the columns in tables A133232 and A133233 are given by this sequence via the formula: if n < k + k*abs(a(n)) then k, otherwise 1 (1 <= k <= n). - Mats Granvik, Jan 22 2008

Links

Crossrefs

Cf. A014963, A120111, A133232, A133233.

Programs

  • GAP
    Concatenation([1],List([1..70],n->1-Lcm(List([1..n+1],i->i))/Lcm(List([1..n],i->i)))); # Muniru A Asiru, Mar 04 2019
  • Magma
    A120112:= func< n | n eq 0 select 1 else 1-Lcm([1..n+1])/Lcm([1..n]) >;
[A120112(n): n in [0..100]]; // G. C. Greubel, May 05 2023
  • Mathematica
    Table[If[n==0, 1, 1-LCM@@Range[n+1]/(LCM@@Range[n])], {n,0,100}] (* G. C. Greubel, May 05 2023 *)
  • PARI
    a(n) = if (n==0, 1, 1 - lcm(vector(n+1, k, k))/lcm(vector(n, k, k))); \\ Michel Marcus, Sep 11 2016
  • SageMath
    def A120112(n):
    return 1 if n == 0 else 1 - lcm(range(1,n+2)) // lcm(range(1,n+1))
[A120112(n) for n in range(101)] # G. C. Greubel, May 05 2023

Formula

a(n) = 1 - lcm(1,...,n+1)/lcm(1,...,n) for n > 0.
a(n) = 1 - A014963(n+1). - Joerg Arndt, Sep 12 2016

A120108 Number triangle T(n,k) = lcm(1,..,n+1)/lcm(1,..,k+1).

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 12, 6, 2, 1, 60, 30, 10, 5, 1, 60, 30, 10, 5, 1, 1, 420, 210, 70, 35, 7, 7, 1, 840, 420, 140, 70, 14, 14, 2, 1, 2520, 1260, 420, 210, 42, 42, 6, 3, 1, 2520, 1260, 420, 210, 42, 42, 6, 3, 1, 1, 27720, 13860, 4620, 2310, 462, 462, 66, 33, 11, 11, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Examples

			Triangle begins:
    1;
    2,   1;
    6,   3,  1;
   12,   6,  2,  1;
   60,  30, 10,  5, 1;
   60,  30, 10,  5, 1, 1;
  420, 210, 70, 35, 7, 7, 1;

Links

Crossrefs

First column is A003418(n+1). Second column is A025555. Row sums are A120109. Diagonal sums are A120110. Inverse is A120111.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->Lcm(List([1..n+1],i->i))/Lcm(List([1..k+1],i->i))))); # Muniru A Asiru, Feb 26 2019
  • Magma
    A120108:= func< n,k | Lcm([1..n+1])/Lcm([1..k+1]) >;
[A120108(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2023
  • Maple
    T:= (n,k)-> ilcm(seq(q,q=1..n+1))/ilcm(seq(r,r=1..k+1)):
seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Feb 26 2019
  • Mathematica
    f[n_] := f[n] = LCM @@ Range[n];
T[n_, k_] := f[n+1]/f[k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 01 2021 *)
  • SageMath
    def f(n): return lcm(range(1,n+2))
def A120108(n,k):
    return f(n)/f(k)
flatten([[A120108(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 04 2023

Formula

Number triangle T(n,k) = [k<=n]*lcm(1,..,n+1)/lcm(1,..,k+1).

A120113 Bi-diagonal inverse of number triangle A120101.

Original entry on oeis.org

1, -6, 1, 0, -5, 1, 0, 0, -14, 1, 0, 0, 0, -3, 1, 0, 0, 0, 0, -11, 1, 0, 0, 0, 0, 0, -13, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -17, 1, 0, 0, 0, 0, 0, 0, 0, 0, -19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

Subdiagonal is -A120114(n-1).

Examples

			Triangle begins
   1;
  -6,  1;
   0, -5,   1;
   0,  0, -14,  1;
   0,  0,   0, -3,   1;
   0,  0,   0,  0, -11,   1;
   0,  0,   0,  0,   0, -13,  1;
   0,  0,   0,  0,   0,   0, -2,   1;
   0,  0,   0,  0,   0,   0,  0, -17,   1;
   0,  0,   0,  0,   0,   0,  0,   0, -19,  1;
   0,  0,   0,  0,   0,   0,  0,   0,   0, -1,  1;

Links

Crossrefs

Cf. A120101, A120111, A120114.

Programs

  • Magma
    A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
A120113:= func< n,k | k eq n select 1 else k eq n-1 select -A120114(n-1) else 0 >;
[A120113(n,k): k in [0..n], n in [0..16]]; // G. C. Greubel, May 05 2023
  • Mathematica
    A120114[n_]:= LCM@@Range[2*n+4]/(LCM@@Range[2*n+2]);
A120113[n_, k_]:= If[k==n, 1, If[k==n-1, -A120114[n-1], 0]];
Table[A120113[n, k], {n,0,16}, {k,0,n}]//Flatten
  • SageMath
    def A120113(n,k):
    if (kA120113(n,k) for k in range(n+1)] for n in range(17)]) # G. C. Greubel, May 05 2023

Formula

T(n, k) = 1 if k = n, T(n, k) = -A120114(n-1) if k = n-1, otherwise 0. - G. C. Greubel, May 05 2023

A120105 Number triangle T(n,k) = lcm(1,..,2*n+2)/lcm(1,..,2*k+2).

Original entry on oeis.org

1, 6, 1, 30, 5, 1, 420, 70, 14, 1, 1260, 210, 42, 3, 1, 13860, 2310, 462, 33, 11, 1, 180180, 30030, 6006, 429, 143, 13, 1, 360360, 60060, 12012, 858, 286, 26, 2, 1, 6126120, 1021020, 204204, 14586, 4862, 442, 34, 17, 1, 116396280, 19399380, 3879876, 277134, 92378, 8398, 646, 323, 19, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Examples

			Triangle begins:
       1;
       6,     1;
      30,     5,    1;
     420,    70,   14,   1;
    1260,   210,   42,   3,   1;
   13860,  2310,  462,  33,  11,   1;
  180180, 30030, 6006, 429, 143,  13,  1;

Links

Crossrefs

First column is A119634. Second column is A051538. Inverse is A120111.
Cf. A120106, A120107.

Programs

  • GAP
    Flat(List([0..9],n->List([0..n],k->Lcm(List([1..2*n+2],i->i))/Lcm(List([1..2*k+2],i->i))))); # Muniru A Asiru, Feb 26 2019
  • Magma
    [Lcm([1..2*n+2])/Lcm([1..2*k+2]): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2023
  • Maple
    T:= (n,k)-> ilcm(seq(q,q=1..2*n+2))/ilcm(seq(r,r=1..2*k+2)):
seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Feb 26 2019
  • Mathematica
    T[n_, k_]:= LCM@@Range[2*n+2]/(LCM@@Range[2*k+2]);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 04 2023 *)
  • SageMath
    def f(n): return lcm(range(1,2*n+3))
def A120105(n,k):
    return f(n)//f(k)
flatten([[A120105(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 04 2023

Formula

Number triangle T(n,k) = [k<=n] + lcm(1,..,2n+2)/lcm(1,..,2k+2).
From G. C. Greubel, May 04 2023: (Start)
Sum_{k=0..n} T(n, k) = A120106(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A120107(n). (End)
Showing 1-4 of 4 results.

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