

A308628


A Van Ecktype 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
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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 \ / \

./ \ / \ 2 / \ 1 / \ 1 / \ /
/ \ / 0 \ / 0 \ / 0 \ / 1 \ /

\ / \ 3 / \(0)/ \ 2 / \ 1 / \
.\ / \ / 2 \ / 0 \ / 1 \ / \

./ \ / \ 0 / \ 1 / \ 1 / \ /
/ \ / \ / 4 \ / 1 \ / \ /

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 Ecktype sequences for the square and hexagonal lattices end similarly.


LINKS

Table of n, a(n) for n=1..82.


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 Ecktype sequences over the 1D "lattice" and the 2D hexagonal and square lattices, respectively.
Sequence in context: A291878 A131487 A230747 * A181670 A261251 A039991
Adjacent sequences: A308625 A308626 A308627 * A308629 A308630 A308631


KEYWORD

nonn,easy


AUTHOR

Charlie Neder, Jun 11 2019


STATUS

approved



