A370386 Irregular triangle read by rows. An infinite rooted tree having root node 1 in row n = 0. Nodes in row n each have n + 1 children with values m + k, where m is the value of the parent node and k takes the values of all nodes from the root to the parent including the parent itself.

1, 2, 3, 4, 4, 5, 6, 5, 6, 8, 5, 6, 7, 8, 6, 7, 8, 10, 7, 8, 9, 12, 6, 7, 9, 10, 7, 8, 10, 12, 9, 10, 12, 16, 6, 7, 8, 9, 10, 7, 8, 9, 10, 12, 8, 9, 10, 11, 14, 9, 10, 11, 12, 16, 7, 8, 9, 11, 12, 8, 9, 10, 12, 14, 9, 10, 11, 13, 16, 11, 12, 13, 15, 20, 8, 9

Offset

0,2

Comments

The paths through the tree represent integer partitions which contain their own first differences and have least part 1. These partitions are counted, including those with any least part, in A364673.

Links

Table of n, a(n) for n=0..75.

Example

Triangle begins:
  1;
  2;
  3, 4;
  4, 5, 6, 5, 6, 8;
  5, 6, 7, 8, 6, 7, 8, 10, 7, 8, 9, 12, 6, 7, 9, 10, 7, 8, 10, 12, 9, 10, 12, 16;
  ...
The tree starts with root 1 in row n = 0. In row n = 2 the parent node 4 has 3 children using values of k: 1, 2, and 4.
Tree begins:
  row
  [n]
  [0]             1
                  |
  [1]         ____2____
             /         \
  [2]     __3__       __4__
         /  |  \     /  |  \
  [3]   4   5   6   5   6   8

Prog

(Python)
def A370386_rowlist(maxrow):
    A, C = [[(1, )]], [[1]]
    for i in range(maxrow):
        A.append([])
        C.append([])
        for j in A[i]:
            for k in j:
                x = j + (j[-1] + k, )
                A[i+1].append(x)
                C[i+1].append(x[-1])
    return(C)

Crossrefs

The tree if only distinct values are allowed is A114622.

Cf. A000142 (row lengths), A002720 (empirical row sums).

Cf. A364673.

Sequence in context: A248110 A108872 A147847 * A335572 A268680 A126974

Adjacent sequences: A370383 A370384 A370385 * A370387 A370388 A370389

Keyword

nonn,easy,tabf

Author

John Tyler Rascoe, Feb 28 2024

Status

approved