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A308628
A Van Eck-type sequence on the triangular lattice.
0
0, 0, 1, 0, 2, 0, 1, 4, 0, 2, 3, 0, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,5
COMMENTS
For a given lattice, the Van Eck sequence over that lattice is the unique sequence of nonnegative integers such that, if all equal terms are connected by "bridges" that travel between adjacent faces, then each term is the length of the bridge connecting the previous term to a term with lower index, or 0 if no such bridge exists. Generally, the Van Eck sequence of a given lattice is not unique since it depends on the path that the sequence takes through the lattice. This sequence uses a spiral, as in A308625 and A308626, and appears as follows, starting at the cell in parentheses facing upward and traveling clockwise:
--------------------------------
\ / \ / \ 3 / \ 1 / \ / \
.\ / \ / 3 \ / 1 \ / 1 \ / \
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./ \ / \ 2 / \ 1 / \ 1 / \ /
/ \ / 0 \ / 0 \ / 0 \ / 1 \ /
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\ / \ 3 / \(0)/ \ 2 / \ 1 / \
.\ / \ / 2 \ / 0 \ / 1 \ / \
--------------------------------
./ \ / \ 0 / \ 1 / \ 1 / \ /
/ \ / \ / 4 \ / 1 \ / \ /
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Note: This sequence uses the definition that two cells are adjacent if they share an edge. Allowing vertex adjacency makes a very boring sequence: 0, 0, 1, and 0, followed by an infinite string of 1's.
a(n) = 1 for all n >= 17, since the previous 1 will always be adjacent to another 1. The Van Eck-type sequences for the square and hexagonal lattices end similarly.
FORMULA
G.f.: x^3*(1 - x + 2*x^2 - 2*x^3 + x^4 + 3*x^5 - 4*x^6 + 2*x^7 + x^8 - 3*x^9 + 2*x^10 + x^11 - 2*x^13)/(1 - x). - Elmo R. Oliveira, Aug 03 2024
EXAMPLE
a(7) = 1, and the only other 1 to appear so far is 4 cells away (not 2, since we only consider edge adjacency), so a(8) = 4.
CROSSREFS
Cf. A181391, A308625, and A308626 for Van Eck-type sequences over the 1D "lattice" and the 2D hexagonal and square lattices, respectively.
Sequence in context: A291878 A131487 A230747 * A181670 A261251 A341101
KEYWORD
nonn,easy
AUTHOR
Charlie Neder, Jun 11 2019
STATUS
approved