A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

1, 6, 0, 6, 66, 0, 12, 0, 150, 0, 30, 1020, 0, 420, 0, 84, 0, 6, 0, 3444, 0, 1302, 0, 252, 0, 42, 19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24, 0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18, 449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30

Offset

0,2

Comments

Rows 0 and 2 have 1 element each; row 1 is empty; for n > 2, we have 0 <= k <= A069813(n).

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002898.

Links

Table of n, a(n) for n=0..67.

Example

The triangle begins:
[1]
[]
[6]
[0, 6]
[66, 0, 12]
[0, 150, 0, 30]
[1020, 0, 420, 0, 84, 0, 6]
[0, 3444, 0, 1302, 0, 252, 0, 42]
[19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24]
[0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18]
[449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30]
...

Crossrefs

Cf. A069813 (greatest area), A002898 (all closed walks), A352838 (square lattice).

For n > 1, row n seems to end with A109047(n).

Sequence in context: A222722 A222747 A222608 * A021900 A351401 A273413

Adjacent sequences: A353088 A353089 A353090 * A353092 A353093 A353094

Keyword

nonn,tabf,walk

Author

Andrey Zabolotskiy, Apr 22 2022

Status

approved