A364038 Triangle read by rows: n-th row gives the number of pieces of size 1 X k required to build all free n-ominoes simultaneously, 1 <= k <= n. Long pieces are preferred over short ones.

1, 0, 1, 1, 1, 1, 2, 4, 2, 1, 11, 6, 8, 2, 1, 35, 21, 20, 13, 3, 1, 118, 83, 60, 43, 19, 3, 1, 461, 312, 202, 175, 75, 25, 4, 1, 1800, 1198, 772, 605, 339, 111, 33, 4, 1, 7114, 4822, 2992, 2248, 1386, 557, 167, 41, 5, 1, 28327, 19291, 12002, 8335, 5638, 2473, 880, 230, 51, 5, 1

Offset

1,7

Comments

More precisely, the preference for long pieces means that the n-th row is the colexicographically largest n-tuple (x_1, ..., x_n) such that all n-ominoes can be built by x_k pieces of size 1 X k, 1 <= k <= n (with no pieces left over). - Pontus von Brömssen, Aug 09 2023

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..136 (rows 1..16)

Michel M. Dauchez, Wood sections used for polyominoes.

Formula

Sum_{k=1..n} k*T(n,k) = n*A000105(n). - Pontus von Brömssen, Aug 06 2023

Example

Trominoes (n=3) are made with 3 and 2 + 1 sections, so row 3 is (1,1,1).
Pentominoes (n=5) are 5, 2(4+1), 4(3+2), 4(3+1+1) and 2+2+1 (sections ordered giving 11, 6, 8, 2, 1 as row 5).
The table begins as follows:
   1;
   0, 1;
   1, 1, 1;
   2, 4, 2, 1;
  11, 6, 8, 2, 1;
  ...
The table below shows how the polyomino shapes (up to pentominoes) are characterized for purposes of this sequence:
.
  polyomino           #squares
    class     shape    per row   characterization
  -----------------------------------------------
  monomino    X           1      "1"
  -----------------------------------------------
  domino      XX          2      "2"
  -----------------------------------------------
  tromino     XXX         3      "3"
.
              XX          2      "2 + 1"
              X           1
  -----------------------------------------------
  tetromino   XXXX        4      "4"
.
              XXX         3      "3+1"
              X           1
.
              XXX         3      "3+1"
               X          1
.
              XX          2      "2+2"
               XX         2
.
              XX          2      "2+2"
              XX          2
  -----------------------------------------------
  pentomino   XXXXX       5      "5"
.
              XXXX        4      "4+1"
              X           1
.
              XXXX        4      "4+1"
               X          1
.
              XXX         3      "3+2"
               XX         2
.
              XXX         3      "3+2"
                XX        2
.
              XXX         3      "3+1+1"
              X X         2
.
              XXX         3      "3+2"
              X           1
              X           1
.
              XXX         3      "3+2"
               X          1
               X          1
.
              X           1      "3+1+1"
              XXX         3
               X          1
.
              X           1      "3+1+1"
              XXX         3
                X         1
.
               X          1      "3+1+1"
              XXX         3
               X          1
.
              XX          2      "2+2+1"
               XX         2
                X         1

Crossrefs

Cf. A000105.

Sequence in context: A303118 A037178 A352650 * A328192 A077748 A294389

Adjacent sequences: A364035 A364036 A364037 * A364039 A364040 A364041

Keyword

nonn,tabl

Author

Michel M. Dauchez, Jul 02 2023

Extensions

Rows 6-10 and name clarified by Pontus von Brömssen, Aug 09 2023

Row 11 from Pontus von Brömssen, Jan 17 2025

Status

approved