A206345 -id:A206345 - cp's OEIS frontend

A206344 a(n) = floor(n/2)^n.

0, 1, 1, 16, 32, 729, 2187, 65536, 262144, 9765625, 48828125, 2176782336, 13060694016, 678223072849, 4747561509943, 281474976710656, 2251799813685248, 150094635296999121, 1350851717672992089, 100000000000000000000, 1000000000000000000000, 81402749386839761113321

Offset: 1

Nathaniel Johnston, Feb 06 2012

The sequence gives the number of (potentially unsolvable) "clock puzzles" with n positions in the video game Final Fantasy XIII-2.
Functions from [n] to [n] with f(i) even or f(i) = 1 for all i. - Olivier Gérard, Sep 23 2016
AGM transform of A059841. See A368366 for the definition of the AGM transform. - Alois P. Heinz, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 1..425
N. Johnston, Counting and Solving Final Fantasy XIII-2's Clock Puzzles

Cf. A206345, A206346, A276978, A276979 (other classes of endofunctions defined by image parity).

Cf. A059841, A368366.

[Floor(n/2)^n: n in [1..30]]; // G. C. Greubel, Mar 31 2023

Maple
```
seq(floor(n/2)^n, n=1..50);
```
Mathematica
```
Table[Floor[n/2]^n, {n,30}]
```

[(n//2)^n for n in range(1,31)] # G. C. Greubel, Mar 31 2023

A206346 Number of solvable clock puzzles with n positions in Final Fantasy XIII-2, up to rotation and reflection.

Original entry on oeis.org

0, 1, 1, 4, 8, 58, 177, 2196, 11091, 172522, 1239350, 20916154, 197149146

Offset: 1

Nathaniel Johnston, Feb 06 2012

Equals the number of Hamiltonian directed graphs on n vertices with the properties that: (1) every vertex has outdegree 1 or 2; and (2) the vertices can be arranged in a circle so that the directed edges leaving each vertex are symmetric about that vertex (e.g., if there is a directed edge that points two vertices in the clockwise direction, then the other one must point two vertices in the counterclockwise direction).

The same as A206345, except clock puzzles that are simply rotations or reflections of each other are not counted multiple times.

Nathaniel Johnston, Counting and Solving Final Fantasy XIII-2's Clock Puzzles.

Cf. A206344, A206345.

a(6)-a(9) corrected, a(10)-a(13) added by Max Alekseyev, Jul 31 2025

cp's OEIS Frontend

A206344 a(n) = floor(n/2)^n.

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