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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

%I #7 Sep 17 2020 12:14:20

%S 1,7,13,26,43,62

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

%C 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.

%C o

%C / \

%C o - o

%C / \ / \

%C j+1 ------ o - O - O

%C / \ / \ / \

%C j ---- o - O- i,j -O

%C / \ / \ / \ / \

%C j-1 -- o - o - O - O - o

%C / \ / \ / \ / \ / \

%C o - o - o - o - o - o

%C / / /

%C i-1 i i+1

%C 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).

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

%H IBM Research, <a href="https://www.research.ibm.com/haifa/ponderthis/challenges/December2012.html">Maximal sum 6x6 grid</a>, Ponder This December 2012.

%e a(1) = 1 for the degenerate triangle.

%e a(2) = 7:

%e 1

%e / \

%e 2 - 3

%e and the equivalent figures resulting from rotation and reflection.

%e .

%e a(3) = 13: 1 solution; the shown versions are all equivalent.

%e 1 1 2 2 2 2

%e / \ / \ / \ / \ / \ / \

%e 3 - 4 4 - 3 1 - 3 1 - 4 3 - 1 4 - 1

%e / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \

%e 2 - 1 - 2 2 - 1 - 2 2 - 4 - 1 2 - 3 - 1 1 - 4 - 2 1 - 3 - 2

%e .

%e a(4) = 26: 6 essentially distinct solutions

%e 1 1 - 4 - 2 - 3 1 1 - 5 - 3 - 2 1 1 - 4 - 3 - 2

%e / \ \ / \ / \ / / \ \ / \ / \ / / \ \ / \ / \ /

%e 2 - 4 5 - 3 - 1 3 - 4 4 - 2 - 1 3 - 4 5 - 2 - 1

%e / \ / \ \ / \ / / \ / \ \ / \ / / \ / \ \ / \ /

%e 4 - 3 - 1 2 - 4 1 - 2 - 5 3 - 4 4 - 2 - 5 3 - 4

%e / \ / \ / \ \ / / \ / \ / \ \ / / \ / \ / \ \ /

%e 1 - 5 - 2 - 3 1 2 - 4 - 3 - 1 1 2 - 1 - 3 - 1 1

%e .

%e a(5) = 43: 4 essentially distinct solutions

%e 2 2 - 1 - 5 - 2 - 3 3 3 - 1 - 4 - 2 - 3

%e / \ \ / \ / \ / \ / / \ \ / \ / \ / \ /

%e 1 - 3 4 - 3 - 4 - 1 1 - 2 2 - 5 - 3 - 1

%e / \ / \ \ / \ / \ / / \ / \ \ / \ / \ /

%e 4 - 2 - 5 5 - 2 - 5 3 - 5 - 4 3 - 6 - 4

%e / \ / \ / \ \ / \ / / \ / \ / \ \ / \ /

%e 5 - 3 - 4 - 1 1 - 3 2 - 6 - 3 - 1 1 - 2

%e / \ / \ / \ / \ \ / / \ / \ / \ / \ \ /

%e 2 - 1 - 5 - 2 - 3 2 3 - 1 - 4 - 2 - 3 3

%e .

%e a(6) = 62: 4 essentially distinct solutions

%e 1 2 - 1 - 2 - 3 - 1 - 3 1 3 - 1 - 3 - 2 - 1 - 2

%e / \ \ / \ / \ / \ / \ / / \ \ / \ / \ / \ / \ /

%e 4 - 5 3 - 4 - 5 - 6 - 2 4 - 5 2 - 6 - 5 - 4 - 3

%e / \ / \ \ / \ / \ / \ / / \ / \ \ / \ / \ / \ /

%e 3 - 2 - 3 1 - 6 - 1 - 4 3 - 2 - 3 4 - 1 - 6 - 1

%e / \ / \ / \ \ / \ / \ / / \ / \ / \ \ / \ / \ /

%e 1 - 4 - 1 - 4 3 - 2 - 3 4 - 1 - 4 - 1 3 - 2 - 3

%e / \ / \ / \ / \ \ / \ / / \ / \ / \ / \ \ / \ /

%e 3 - 6 - 5 - 6 - 2 4 - 5 2 - 6 - 5 - 6 - 3 4 - 5

%e / \ / \ / \ / \ / \ \ / / \ / \ / \ / \ / \ \ /

%e 2 - 1 - 2 - 3 - 1 - 3 1 3 - 1 - 3 - 2 - 1 - 2 1

%Y Cf. A000217, A003215, A228882.

%K nonn,hard,more

%O 1,2

%A _Hugo Pfoertner_, Sep 16 2020