A372264 a(n) = n! - n^2 + 2n - 1.

1, 1, 2, 15, 104, 695, 5004, 40271, 362816, 3628719, 39916700, 479001479, 6227020656, 87178291031, 1307674367804, 20922789887775, 355687428095744, 6402373705727711, 121645100408831676, 2432902008176639639, 51090942171709439600, 1124000727777607679559, 25852016738884976639516

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

The number of distinct cards in a deck that has each card twice to perform the n-card trick, where the assistant chooses the hidden card. This number corresponds to a particular strategy, where if there is a duplicate card, then the assistant puts one of those duplicates on the far left and hides the other. The assistant then arranges all the other cards in nondecreasing order. If there are no duplicates, the strategy is similar to the standard one.

Links

• Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 19.
• Michael Kleber and Ravi Vakil, The best card trick, The Mathematical Intelligencer 24 (2002), 9-11.

Crossrefs

Cf. A370888, A371217, A372255, A372256.

Programs

Mathematica

Table[(n! - n^2 + 2 n - 1), {n, 1, 25}]

Python

from math import factorial
def A372264(n): return factorial(n)-(n-1)**2 # Chai Wah Wu, May 02 2024

Formula

E.g.f.: 1/(1 - x) - exp(x)*(1 - x + x^2). - Stefano Spezia, Jun 06 2024