A337765 Number of addition triangles with apex n where all rows are weakly increasing.

1, 2, 2, 4, 4, 5, 6, 9, 9, 11, 12, 15, 16, 18, 20, 26, 27, 29, 32, 37, 39, 43, 47, 53, 55, 60, 65, 72, 75, 80, 88, 99, 102, 108, 114, 125, 132, 141, 148, 159, 166, 176, 187, 200, 206, 218, 232, 249, 257, 268, 282, 301, 313, 327, 340, 360, 374, 393, 410, 429, 444, 465, 487, 516, 530, 550

Offset: 1

Author

Seiichi Manyama, Sep 19 2020

Keywords

• nonn

Comments

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

If the bottom row are weakly increasing, then every rows are weakly increasing.

5

2<=3

1<=1<=2

Examples

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

Links

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

Crossrefs

Cf. A062684, A062896, A337766.

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.sort
      }
    }
    f_ary = b_ary
    cnt += f_ary.size
  end
  cnt
end
def A337765(n)
  (1..n).map{|i| A(i)}
end
p A337765(50)