A372256 a(n) = (n-1)!/2^floor((n-1)/2) + floor((n-1)/2).

1, 1, 2, 4, 8, 32, 93, 633, 2524, 22684, 113405, 1247405, 7484406, 97297206, 681080407, 10216206007, 81729648008, 1389404016008, 12504636144009, 237588086736009, 2375880867360010, 49893498214560010, 548828480360160011, 12623055048283680011, 151476660579404160012

Offset: 1

Author

Tanya Khovanova and the MIT PRIMES STEP junior group, Apr 24 2024

Keywords

• nonn

Comments

The maximum number of distinct cards in a deck that has each card twice to perform the n-card trick, where the audience chooses the hidden card.

Examples

			Consider a five-card trick, where the assistant gets four cards from a deck and is told which card to hide. Moreover, the deck has a duplicate of each card. In the worst case scenario, the assistant gets two duplicates and has to hide the other card. There are six different ways to arrange two pairs of cards. Thus, the assistant can signal a number 1 through 6. The hidden card can't take a value of the cards on the table, so the maximum number of distinct values is 8. Thus a(5) = 8.
		

Links

• Michael Kleber and Ravi Vakil, The best card trick, The Mathematical Intelligencer 24 (2002), 9-11.
• Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, Samuel Tsui, and Tanya Khovanova, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024.

Crossrefs

Cf. A370888, A371217, A372255.

Programs

Mathematica

Table[(K - 1) !/(2^Floor[(K - 1)/2]) + Floor[(K - 1)/2], {K, 1, 25}]