A030495 a(n) = (n+1)! + n.

1, 3, 8, 27, 124, 725, 5046, 40327, 362888, 3628809, 39916810, 479001611, 6227020812, 87178291213, 1307674368014, 20922789888015, 355687428096016, 6402373705728017, 121645100408832018, 2432902008176640019, 51090942171709440020, 1124000727777607680021, 25852016738884976640022

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Numbers m such that n!*C(m,n) = C(m,n+1). - Lekraj Beedassy, Feb 18 2006

a(n) is also the maximum size for a deck of cards in the Communicating the Card magic trick. In this game Alice draws n+1 cards from the deck at random, without replacement, and passes n of them, one by one, to her accomplice Bob. If the deck has a(n) cards or fewer, there is an algorithm by which Alice can communicate to Bob the identity of the card she chooses to retain, using only the identity and the order of passing of the n passed cards. (One side of the proof, that no larger deck size will work, is easy: the retained card can be one of (n+1)! possibilities, since Bob knows that it is not one of the n passed cards. Alice has (n+1) ways to retain a card and n! ways to order the passing of the remaining cards, so she cannot communicate more than (n+1)! different possibilities.) - Lee A. Newberg, Jun 09 2010

Examples

			a(5) = (5+1)!+5 = 725.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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. 7.
• Pr. Malik Magdon-Ismail, Communicating the Card puzzle.
• Shai Simonson and Tara S. Holm, Using A Card Trick To Teach Discrete Mathematics, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Volume 13, Issue 3, 2003, DOI:10.1080/10511970308984061.

Crossrefs

a(n) = least k such that s(k) = n, where s=A030298.

Equals A005095(n+1) - 1.

Programs

Magma

[Factorial(n+1)+ n: n in [0..30]]; // Vincenzo Librandi, Feb 04 2013

Maple

A030495:=n->(n+1)! + n; seq(A030495(n), n=0..40); # Wesley Ivan Hurt, Mar 04 2014

Mathematica

Table[(n+1)!+n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

Formula

a(n) = n + Sum_{k=1..n-1} k*k!.

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

Extensions

Better description from Jason Earls, Mar 24 2001