A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.

1, 2, 3, 3, 4, 3, 5, 5, 6, 5, 6, 7, 7, 6, 8, 7, 6, 9, 9, 9, 10, 9, 9, 10, 11, 11, 9, 10, 12, 11, 12, 10, 13, 13, 12, 10, 14, 13, 12, 14, 15, 15, 15, 14, 15, 16, 15, 15, 14, 15, 17, 17, 15, 14, 15, 18, 17, 18, 18, 15, 19, 19, 18, 18, 15, 20, 19, 18, 18, 20, 21, 21, 21, 18, 20, 21, 22, 21, 21, 22, 20, 21

Offset

1,2

Comments

T(n,k) is also the positive integer whose partition into k consecutive parts is associated to the k-th vertex, from left to right, of the largest Dyck path of the symmetric representation of sigma(n). For more information see A237593.

Also this triangle can be constructed replacing every zero of triangle A285891 with the previous positive integer from the same column.

Links

Table of n, a(n) for n=1..82.

Example

Triangle begins:
   1;
   2;
   3,  3;
   4,  3;
   5,  5;
   6,  5,  6;
   7,  7,  6;
   8,  7,  6;
   9,  9,  9;
  10,  9,  9, 10;
  11, 11,  9, 10;
  12, 11, 12, 10;
  13, 13, 12, 10;
  14, 13, 12, 14;
  15, 15, 15, 14, 15;
  16, 15, 15, 14, 15;
  17, 17, 15, 14, 15;
  18, 17, 18, 18, 15;
  19, 19, 18, 18, 15;
  20, 19, 18, 18, 20;
  21, 21, 21, 18, 20, 21;
  22, 21, 21, 22, 20, 21;
  23, 23, 21, 22, 20, 21;
  24, 23, 24, 22, 20, 21;
  25, 25, 24, 22, 25, 21;
  26, 25, 24, 26, 25, 21;
  27, 27, 27, 26, 25, 27;
  28, 27, 27, 26, 25, 27, 28;
...

Crossrefs

Column k stars with A000217(k) in the row A000217(k).

Row n has length A003056(n).

Cf. A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

Sequence in context: A255239 A316339 A186971 * A262535 A096827 A298321

Adjacent sequences: A329252 A329253 A329254 * A329256 A329257 A329258

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 09 2019

Status

approved