A337433 Maximum sum of node values defined in analogy to A228882 for a triangular region of the hexagonal lattice with n*(n+1)/2 points.

1, 7, 13, 26, 43, 62

Offset

1,2

Comments

In a triangle of n*(n+1)/2 grid points of the hexagonal lattice, the grid points are assigned integers b(j,k) > 0, such that if b(j,k) = i, then all numbers 1 ... i-1 are represented in the node values of the nearest neighbors (i-1,j), (i-1,j+1), (i,j-1), (i,j+1), (i+1,j-1), (i+1,j) of point (i,j) in the lattice.

o

/ \

o - o

/ \ / \

j+1 ------ o - O - O

/ \ / \ / \

j ---- o - O- i,j -O

/ \ / \ / \ / \

j-1 -- o - o - O - O - o

/ \ / \ / \ / \ / \

o - o - o - o - o - o

/ / /

i-1 i i+1

In the interior of the figure, there are A003215(1)-1 = 6 nearest neighbors (hence b(j,k) <= 7); points on the sides of the triangle have 4 nearest neighbors (b(j,k) <= 5), and the corners of the triangle have 2 nearest neighbors (b(j,k) <= 3).

a(n) is the sum of the n*(n+1)/2 = A000217(n) values of b(j,k).

Links

Table of n, a(n) for n=1..6.

IBM Research, Maximal sum 6x6 grid, Ponder This December 2012.

Example

a(1) = 1 for the degenerate triangle.
a(2) = 7:
    1
   / \
  2 - 3
and the equivalent figures resulting from rotation and reflection.
.
a(3) = 13: 1 solution; the shown versions are all equivalent.
      1          1          2          2          2          2
     / \        / \        / \        / \        / \        / \
    3 - 4      4 - 3      1 - 3      1 - 4      3 - 1      4 - 1
   / \ / \    / \ / \    / \ / \    / \ / \    / \ / \    / \ / \
  2 - 1 - 2  2 - 1 - 2  2 - 4 - 1  2 - 3 - 1  1 - 4 - 2  1 - 3 - 2
.
a(4) = 26: 6 essentially distinct solutions
        1    1 - 4 - 2 - 3    1    1 - 5 - 3 - 2    1    1 - 4 - 3 - 2
       / \    \ / \ / \ /    / \    \ / \ / \ /    / \    \ / \ / \ /
      2 - 4    5 - 3 - 1    3 - 4    4 - 2 - 1    3 - 4    5 - 2 - 1
     / \ / \    \ / \ /    / \ / \    \ / \ /    / \ / \    \ / \ /
    4 - 3 - 1    2 - 4    1 - 2 - 5    3 - 4    4 - 2 - 5    3 - 4
   / \ / \ / \    \ /    / \ / \ / \    \ /    / \ / \ / \    \ /
  1 - 5 - 2 - 3    1    2 - 4 - 3 - 1    1    2 - 1 - 3 - 1    1
.
a(5) = 43: 4 essentially distinct solutions
          2     2 - 1 - 5 - 2 - 3     3     3 - 1 - 4 - 2 - 3
         / \     \ / \ / \ / \ /     / \     \ / \ / \ / \ /
        1 - 3     4 - 3 - 4 - 1     1 - 2     2 - 5 - 3 - 1
       / \ / \     \ / \ / \ /     / \ / \     \ / \ / \ /
      4 - 2 - 5     5 - 2 - 5     3 - 5 - 4     3 - 6 - 4
     / \ / \ / \     \ / \ /     / \ / \ / \     \ / \ /
    5 - 3 - 4 - 1     1 - 3     2 - 6 - 3 - 1     1 - 2
   / \ / \ / \ / \     \ /     / \ / \ / \ / \     \ /
  2 - 1 - 5 - 2 - 3     2     3 - 1 - 4 - 2 - 3     3
.
a(6) = 62: 4 essentially distinct solutions
            1     2 - 1 - 2 - 3 - 1 - 3     1     3 - 1 - 3 - 2 - 1 - 2
           / \     \ / \ / \ / \ / \ /     / \     \ / \ / \ / \ / \ /
          4 - 5     3 - 4 - 5 - 6 - 2     4 - 5     2 - 6 - 5 - 4 - 3
         / \ / \     \ / \ / \ / \ /     / \ / \     \ / \ / \ / \ /
        3 - 2 - 3     1 - 6 - 1 - 4     3 - 2 - 3     4 - 1 - 6 - 1
       / \ / \ / \     \ / \ / \ /     / \ / \ / \     \ / \ / \ /
      1 - 4 - 1 - 4     3 - 2 - 3     4 - 1 - 4 - 1     3 - 2 - 3
     / \ / \ / \ / \     \ / \ /     / \ / \ / \ / \     \ / \ /
    3 - 6 - 5 - 6 - 2     4 - 5     2 - 6 - 5 - 6 - 3     4 - 5
   / \ / \ / \ / \ / \     \ /     / \ / \ / \ / \ / \     \ /
  2 - 1 - 2 - 3 - 1 - 3     1     3 - 1 - 3 - 2 - 1 - 2     1

Crossrefs

Cf. A000217, A003215, A228882.

Sequence in context: A060455 A205541 A072579 * A067870 A147258 A146718

Adjacent sequences: A337430 A337431 A337432 * A337434 A337435 A337436

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Sep 16 2020

Status

approved