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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
1, 7, 13, 26, 43, 62 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 30 15:50 EDT 2022. Contains 357106 sequences. (Running on oeis4.)