A206346 -id:A206346 - 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

A206345 Number of solvable clock puzzles with n positions in Final Fantasy XIII-2.

Original entry on oeis.org

0, 1, 1, 13, 32, 507, 1998, 33136, 193995, 3426518, 27187715, 501668052, 5124675822

Offset: 1

Nathaniel Johnston, Feb 06 2012

The sequence gives the number of ways of placing the integers 1, 2, ..., floor(n/2) (with repetition) in n spaces on a circle so that you can jump to every integer exactly once, and the distance you jump is equal to the integer you are currently standing on.
A206344 is a trivial upper bound.
This is the same as A206346, except clock puzzles that are rotations or reflections of each other are counted as distinct.

			A solvable clock puzzle in the n = 6 case arises from the following integers (placed clockwise around a circle): 1, 3, 3, 2, 1, 3. If we label the positions 0, 1, 2, 3, 4, 5, then a solution to this puzzle is the following sequence of positions: 0, 1, 4, 3, 5, 2.

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

Cf. A206344, A206346.

a(10) from Nathaniel Johnston, Feb 07 2012

a(11)-a(13) from Bert Dobbelaere, Apr 28 2021

cp's OEIS Frontend

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

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