A387494 Array read by antidiagonals: T(n,k) is the maximum number of steps in a n^k cuboid with a rule that allows moving to neighboring visited cuboids if the next step is onto an unvisited cuboid when starting in a corner.

0, 0, 1, 0, 4, 2, 0, 11, 12, 3, 0, 26, 46, 24, 4

Offset

1,5

Comments

There are 2 step patterns, those being: unvisited -> visited -> unvisited, unvisited -> unvisited.

Links

Table of n, a(n) for n=1..15.

Formula

T(1, k) = 0.

T(n, 1) = n - 1.

T(n, 2) = 2*n*(n-1) = A046092(n-1).

Conjecture for n > 1: T(n, 3) = 2*n^3 - 3*n + floor(3/n).

Conjecture for general T(n, k) where delta(P) is the Kronecker delta (1 if P is true, 0 if P is false): delta(k != 0)*delta(n > 1)*(2*n^k - k*n + floor(k/n) - delta(k == 1) - delta(n == 2)*delta(k == 2)).

Example

Table for T(n, k):
  n/k |  1     2    3    4
------+-----------------------
  1   |  0     0    0    0 ...
  2   |  1     4   11   26 ...
  3   |  2    12   46  ...
  4   |  3    24  ...
  5   |  4   ...
    ...
For n = 2 and k = 2, a solution with T(2, 2) = 4 steps is (0, 0) -> (0, 1) -> (0, 0) -> (1, 0) -> (1, 1). The initial (0,0) is the starting coordinate and (0,1) is the first step. No solutions are possible with more than 4 steps.
T(2,3) = 11: 000 -> 001 -> 000 -> 010 -> 000 -> 100 -> 101 -> 001 -> 011 -> 010 -> 110 -> 111.
T(3,2) = 12: 00 -> 01 -> 00 -> 10 -> 11 -> 01 -> 02 -> 12 -> 11 -> 21 -> 20 -> 21 -> 22.
From Andrew Howroyd, Sep 01 2025: (Start)
T(3,3) = 46: 000 -> 001 -> 002 -> 001 -> 011 -> 111 -> 110 -> 111 -> 112 -> 111 -> 101 -> 111 -> 121 -> 111 -> 211 -> 201 -> 101 -> 100 -> 000 -> 010 -> 011 -> 012 -> 002 -> 102 -> 112 -> 122 -> 112 -> 212 -> 211 -> 210 -> 110 -> 120 -> 121 -> 021 -> 121 -> 221 -> 220 -> 120 -> 020 -> 021 -> 022 -> 122 -> 222 -> 212 -> 202 -> 201 -> 200. (End)

Crossrefs

Columns k=1..2 are A001477(n-1), A046092(n-1).

Sequence in context: A091435 A330472 A396351 * A118441 A244131 A206428

Adjacent sequences: A387491 A387492 A387493 * A387495 A387496 A387497

Keyword

nonn,tabl,more

Author

Ben Samberg, Aug 31 2025

Extensions

T(2,4) and T(3,3) from Andrew Howroyd, Aug 31 2025

Status

approved