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

1,3

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

Table of n, a(n) for n=1..23.

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.

Formula

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

Mathematica

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

Prog

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

Crossrefs

Cf. A370888, A371217, A372255, A372256.

Sequence in context: A027080 A208347 A293045 * A081018 A006675 A215643

Adjacent sequences: A372261 A372262 A372263 * A372265 A372266 A372267

Keyword

nonn,easy

Author

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

Status

approved