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.

A382252 Triangle T(n,k) = numerator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 1, 5, 3, 4, 1, 2, 7, 8, 5, 1, 7, 1, 3, 5, 6, 1, 8, 9, 2, 11, 12, 7, 1, 3, 5, 11, 1, 13, 7, 8, 1, 10, 11, 4, 13, 2, 5, 16, 9, 1, 11, 3, 13, 7, 3, 1, 17, 9, 10, 1, 4, 13, 14, 5, 16, 17, 2, 19, 20, 11, 1, 13, 7, 1, 2, 17, 3, 19, 1, 7, 11, 12, 1, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24
Offset: 0

Views

Author

M. F. Hasler, Apr 15 2025

Keywords

Comments

Since the operation n @ k := (n + k)/(1 + n*k) is commutative, it is sufficient to list only the lower half of the "multiplication table", which would otherwise be an infinite square array. This triangle lists the numerators, and A382253 lists the denominators.

Examples

			The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows:
(0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore row and column 0 give the column and row headers.)
  0    1    2     3     4     5     6     7     8    Numerators of  0;
  1    1    1     1     1     1     1     1     1     lower left    1, 1:
  2    1   4/5   5/7   2/3   7/11  8/13  3/5  10/17    triangle:    2, 1, 4;
  3    1   5/7   3/5   7/13  1/2   9/19  5/11 11/25                 3, 1, 5, 3
  4    1   2/3   7/13  8/17  3/7   2/5  11/29  4/11                 4, 1, 2, 7, 8;
  5    1   7/11  1/2   3/7   5/13 11/31  1/3  13/41                 etc.
  6    1   8/13  9/19  2/5  11/31 12/37 13/43  2/7
  7    1   3/5   5/11 11/29  1/3  13/43  7/25  5/19
  8    1  10/17 11/25  4/11 13/41  2/7   5/19 16/65
This sequence lists the numerators of the values, where numerator(x) = x for integers, and only for the lower left triangle of the table, by rows.

Crossrefs

Cf. A382253 (denominators), A382257 (related); A228564 (main diagonal), A001477 (row & col. 0), A000012 (row & col. 1).

Programs

  • PARI
    apply( {A382252(n,k=-1)= k<0&& k=n-(1+n=(sqrtint(8*n+1)-1)\2)*n/2; numerator((n+k)/(1+n*k))}, [0..30])

Formula

T(n,k) = T(k,n) for all n, k >= 0; therefore only k <= n is considered here.
T(n,0) = T(0,n) = n and T(n,1) = T(1,n) = 1 for all n >= 0.
T(n,n) = A022998(n) = n if odd, else 2*n.

A382253 Triangle T(n,k) = denominator of (n+k)/(1+n*k), 0 <= k <= n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 7, 5, 1, 1, 3, 13, 17, 1, 1, 11, 2, 7, 13, 1, 1, 13, 19, 5, 31, 37, 1, 1, 5, 11, 29, 3, 43, 25, 1, 1, 17, 25, 11, 41, 7, 19, 65, 1, 1, 19, 7, 37, 23, 11, 4, 73, 41, 1, 1, 7, 31, 41, 17, 61, 71, 9, 91, 101, 1
Offset: 0

Views

Author

M. F. Hasler, Apr 15 2025

Keywords

Comments

Since the operation n @ k := (n + k)/(1 + n*k) is commutative, it is sufficient to list only the lower half of the "multiplication table", which would otherwise be an infinite square array. This triangle lists the denominators, and A382252 lists the numerators.

Examples

			The table for the operation n @ k := (n + k)/(1 + n*k) starts as follows:
(0 is the neutral element for the operation: n @ 0 = n = 0 @ n, therefore the elements in row 0 and column 0 equal the column and row index.)
  0    1    2     3     4     5     6     7     8    Denominators of lower left
  1    1    1     1     1     1     1     1     1    triangle:  1;
  2    1   4/5   5/7   2/3   7/11  8/13  3/5  10/17             1, 1
  3    1   5/7   3/5   7/13  1/2   9/19  5/11 11/25             1, 1, 5;
  4    1   2/3   7/13  8/17  3/7   2/5  11/29  4/11             1, 1, 7, 5;
  5    1   7/11  1/2   3/7   5/13 11/31  1/3  13/41             1, 1, 3, 13, 17;
  6    1   8/13  9/19  2/5  11/31 12/37 13/43  2/7                 etc.
  7    1   3/5   5/11 11/29  1/3  13/43  7/25  5/19
  8    1  10/17 11/25  4/11 13/41  2/7   5/19 16/65
The sequence lists the denominators of the values, where denominator(x) = 1 for integers, and only for the lower left triangle of the table, by rows.

Crossrefs

Cf. A382252, A382257; A228564 (main diagonal), A001477 (row & col. 0 of the '@' table), A000012 (row & col. 0 & 1 of the table of denominators).

Programs

  • PARI
    apply( {A382253(n,k=-1)= k<0&& k=n-(1+n=(sqrtint(8*n+1)-1)\2)*n/2; denominator((n+k)/(1+n*k))}, [0..66])

Formula

T(n,k) = T(k,n) for all n, k >= 0;
T(n,0) = T(0,n) = T(n,1) = T(1,n) = 1 for all n >= 0;
T(n,n) = denominator(2*n/(1+n^2)) = numerator((1+n^2)/2) = A228564(n).

A383431 a(n) is the denominator of tanh(Sum_{k=1..n-1} artanh(k/n)), where artanh is the inverse hyperbolic tangent function.

Original entry on oeis.org

1, 2, 11, 18, 127, 463, 1717, 3218, 24311, 92379, 352717, 1352079, 5200301, 20058301, 77558761, 150270098, 1166803111, 4537567651, 17672631901, 68923264411, 269128937221, 1052049481861, 4116715363801, 16123801841551, 63205303218877, 247959266474053, 973469712824057, 3824345300380221, 15033633249770521
Offset: 1

Views

Author

Thomas Ordowski, Apr 27 2025

Keywords

Comments

a(2^m) is even for m > 0.

Examples

			Denominators of 0, 1/2, 9/11, 17/18, 125/127, 461/463, 1715/1717, 3217/3218, ...

Crossrefs

Cf. A001700, A382257 (numerators).

Formula

a(n) = (binomial(2n-1, n-1) + 1)/2 if n = 2^m or a(n) = binomial(2n-1, n-1) + 1 otherwise, because tanh(Sum_{k=1..n-1} artanh(k/n)) = (binomial(2n-1, n-1) - 1)/(binomial(2n-1, n-1) + 1) reduced.
a(n) = A382257(n) + 1 if n = 2^m or a(n) = A382257(n) + 2 otherwise.
Showing 1-3 of 3 results.

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