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

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

1,3

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..500

Example

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

Prog

(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)

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.

Sequence in context: A358636 A347609 A143738 * A029071 A117144 A104408

Adjacent sequences: A337763 A337764 A337765 * A337767 A337768 A337769

Keyword

nonn

Author

Seiichi Manyama, Sep 19 2020

Status

approved