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

Author

Seiichi Manyama, Sep 19 2020

Keywords

• nonn

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

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

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)