A346523 -id:A346523 - cp's OEIS frontend

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

Seiichi Manyama, Sep 19 2020

nonn

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

			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

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

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.

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)

A348850 a(n) is the number of labeled rooted unordered binary trees T where the nodes are labeled with distinct positive integers, the root has label n, each parent label equals the sum of its children labels, and T cannot be extended.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 9, 10, 14, 22, 33, 57, 66, 94, 132, 188, 317, 454, 576, 806, 1083, 1535, 2342, 3215, 5231, 5656, 8545, 10804, 15226, 21153, 30342, 44536, 63165, 73877, 107241, 133994, 178497, 247564, 331695, 472331

Offset: 1

Rémy Sigrist, Nov 01 2021

For any n > 0:
- we can imagine a variant of Grundy's game where we start with n at root position,
- and each move consists in adding to a leaf, say w, two children, u and v such that 0 < u < v and u+v = w and u and v do not already appear in the tree,
- a(n) gives the number of final positions (where no move is possible).

			For n = 1, 2, 3, 4: a(n) = 1:
          |         |         |         |
          1         2         3         4
                             / \       / \
                            1   2     1   3
For n = 5, 6: a(n) = 2:
          |         |         |         |
          5         5         6         6
         / \       / \       / \       / \
        1   4     2   3     1   5     2   4
                               / \       / \
                              2   3     1   3

Rémy Sigrist, PARI program for A348850
Wikipedia, Grundy's game

Cf. A124973, A346523, A348851.

PARI
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See Links section.
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cp's OEIS Frontend

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

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