cp's OEIS Frontend

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

%I A030495 #44 Jun 06 2024 14:10:48
%S A030495 1,3,8,27,124,725,5046,40327,362888,3628809,39916810,479001611,
%T A030495 6227020812,87178291213,1307674368014,20922789888015,355687428096016,
%U A030495 6402373705728017,121645100408832018,2432902008176640019,51090942171709440020,1124000727777607680021,25852016738884976640022
%N A030495 a(n) = (n+1)! + n.
%C A030495 Numbers m such that n!*C(m,n) = C(m,n+1). - _Lekraj Beedassy_, Feb 18 2006
%C A030495 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
%H A030495 Vincenzo Librandi, <a href="/A030495/b030495.txt">Table of n, a(n) for n = 0..200</a>
%H A030495 Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, <a href="https://arxiv.org/abs/2405.21007">Card Tricks and Information</a>, arXiv:2405.21007 [math.HO], 2024. See p. 7.
%H A030495 Pr. Malik Magdon-Ismail, <a href="http://www.cs.rpi.edu/~magdon/miscellaneous/puzzles/puzzles.html">Communicating the Card</a> puzzle.
%H A030495 Shai Simonson and Tara S. Holm, <a href="http://web.stonehill.edu/compsci/cardTrick.pdf">Using A Card Trick To Teach Discrete Mathematics</a>, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Volume 13, Issue 3, 2003, DOI:10.1080/10511970308984061.
%F A030495 a(n) = n + Sum_{k=1..n-1} k*k!.
%F A030495 E.g.f.: 1/(1 - x)^2 + exp(x)*x . - _Stefano Spezia_, Jun 06 2024
%e A030495 a(5) = (5+1)!+5 = 725.
%p A030495 A030495:=n->(n+1)! + n; seq(A030495(n), n=0..40); # _Wesley Ivan Hurt_, Mar 04 2014
%t A030495 Table[(n+1)!+n, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, May 19 2011 *)
%o A030495 (Magma) [Factorial(n+1)+ n: n in [0..30]]; // _Vincenzo Librandi_, Feb 04 2013
%Y A030495 a(n) = least k such that s(k) = n, where s=A030298.
%Y A030495 Equals A005095(n+1) - 1.
%K A030495 nonn
%O A030495 0,2
%A A030495 _Clark Kimberling_
%E A030495 Better description from _Jason Earls_, Mar 24 2001