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.

A062896 Number of addition triangles with apex n (version 2).

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 7, 12, 12, 18, 19, 27, 28, 39, 41, 54, 58, 74, 78, 99, 106, 129, 139, 168, 179, 214, 229, 268, 289, 335, 357, 414, 443, 504, 540, 612, 653, 737, 786, 878, 938, 1045, 1111, 1234, 1313, 1444, 1539, 1692, 1795, 1965, 2082, 2273, 2414
Offset: 1

Views

Author

Naohiro Nomoto, Feb 11 2002

Keywords

Comments

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
Reversing the base does not count as a different triangle.

Examples

			For n = 5:
    5
   2,3     5     5
  1,1,2   4,1   2,3   5.
with four different bases, so a(5) = 4.

Links

Crossrefs

See A062684 for version 1 (counts reversals).
Equivalent sequences with restrictions on rows: A337765 (weakly increasing), A337766 (strongly increasing).
Equivalent sequence where n is the sum of all numbers in the triangle: A337787.
Cf. A028307, A066411.

Extensions

Extended and edited by John W. Layman, Feb 14 2002

A337766 Number of addition triangles with apex n where all rows are strongly increasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 16, 17, 19, 22, 24, 25, 28, 31, 33, 35, 39, 43, 46, 48, 52, 57, 60, 63, 69, 75, 78, 82, 88, 94, 99, 104, 111, 119, 124, 129, 137, 147, 153, 160, 169, 179, 187, 194, 204, 216, 224, 233, 246, 259, 267, 277, 292, 308, 318, 329, 343, 361
Offset: 1

Views

Author

Seiichi Manyama, Sep 19 2020

Keywords

Comments

An addition triangle has any finite sequence of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
If the bottom row is strongly increasing, then every row is strongly increasing.
8
3<5
1<2<3

Examples

			For n = 5:
   5     5
  1,4   2,3   5
For n = 6:
   6     6
  1,5   2,4   6
For n = 7:
   7     7     7
  1,6   2,5   3,4   7
For n = 8:
    8
   3,5     8     8     8
  1,2,3   1,7   2,6   3,5   8
For n = 9:
    9
   3,6     9     9     9     9
  1,2,4   1,8   2,7   3,6   4,5   9

Links

Crossrefs

Equivalent sequences with different restrictions on rows: A062684 (none, except terms are positive), A062896 (not a reversal of a counted row), A337765 (weakly increasing).
Cf. A346523.

Programs

  • Ruby
    def A(n)
  f_ary = [[n]]
  cnt = 1
  while f_ary.size > 0
    b_ary = []
    f_ary.each{|i|
      s = i.size
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        b_ary << a if a.size == s + 1 && a == a.uniq.sort
      }
    }
    f_ary = b_ary
    cnt += f_ary.size
  end
  cnt
end
def A337766(n)
  (1..n).map{|i| A(i)}
end
p A337766(50)

A337785 Number of addition triangles whose sum is n (version 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 9, 1, 9, 4, 9, 3, 14, 2, 14, 6, 14, 5, 21, 4, 19, 10, 21, 8, 27, 6, 29, 16, 25, 12, 38, 14, 33, 19, 37, 22, 46, 14, 47, 33, 45, 22, 59, 29, 59, 35, 56, 40, 74, 34, 68, 53, 72, 47, 90, 47, 88, 63, 88, 64, 105, 59, 108, 84, 106, 75, 130, 81, 125, 99, 128, 103, 147
Offset: 1

Views

Author

Seiichi Manyama, Sep 21 2020

Keywords

Comments

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
Reversing the base counts as a different triangle.

Examples

			   n |
-----+------------------------------------------------
   1 |  1
-----+------------------------------------------------
   2 |  2
-----+------------------------------------------------
   3 |  3
-----+------------------------------------------------
   4 |      2
     |  4  1,1
-----+------------------------------------------------
   5 |  5
-----+------------------------------------------------
   6 |      3    3
     |  6  1,2  2,1
-----+------------------------------------------------
   7 |  7
-----+------------------------------------------------
   8 |      4    4    4
     |  8  1,3  2,2  3,1
-----+------------------------------------------------
   9 |  9
-----+------------------------------------------------
  10 |      5    5    5    5
     | 10  1,4  2,3  3,2  4,1
-----+------------------------------------------------
  11 |       4
     |      2,2
     | 11  1,1,1
-----+------------------------------------------------
  12 |      6    6    6    6    6
     | 12  1,5  2,4  3,3  4,2  5,1
-----+------------------------------------------------
  13 | 13
-----+------------------------------------------------
  14 |                                     5      5
     |      7    7    7    7    7    7    2,3    3,2
     | 14  1,6  2,5  3,4  4,3  5,2  6,1  1,1,2  2,1,1

Links

Crossrefs

Cf. A014430, A062684, A062896, A337765, A337766, see A337787 for version 2.

Programs

  • Ruby
    def f(n)
  ary = [1]
  (n - 1).times{|i|
    ary = [0] + ary + [0]
    ary = (0..i + 1).map{|j| ary[j] + ary[j + 1] + 1}
  }
  ary
end
def A(n)
  f_ary = (1..n / 2).map{|i| [i]}
  cnt = 1
  s = 1
  while f_ary.size > 0
    s_ary = f(s + 1)
    b_ary = []
    f_ary.each{|i|
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        if a.size == s + 1
          sum = (0..s).inject(0){|t, m| t + s_ary[m] * a[m]}
          if sum < n
            b_ary << a
          elsif sum == n
            cnt += 1
          end
        end
      }
    }
    f_ary = b_ary
    s += 1
  end
  cnt
end
def A337785(n)
  (1..n).map{|i| A(i)}
end
p A337785(50)
Showing 1-3 of 3 results.

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