This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371217 #30 Jul 27 2024 17:57:25
%S A371217 1,4,15,52,197,896,4987,33216,257161,2262124,22241671,241476060,
%T A371217 2867551117,36960108680,513753523571,7659705147976,121918431264273,
%U A371217 2063255678027668,36991535865656959,700377953116334788,13963866589144933461,292421219327021540176,6417047546280200867819
%N A371217 The maximum deck size to perform Colm Mulcahy's n-card trick.
%C A371217 With this card trick the magician's assistant gets n cards from a deck, hides one card, and displays the rest, where it is allowed to place some of the displayed cards face down. After that, the magician guesses the hidden card.
%C A371217 The trick for n = 4 was invented by Colm Mulcahy and is a variation of the Fitch Cheney trick. Surprisingly, the largest possible deck is the standard deck of 52 cards.
%D A371217 Wallace Lee, Math Miracles, published by Seeman Printery, Durham, N.C., 1950.
%D A371217 Colm Mulcahy, Mathematical card magic: fifty-two new effects, published by CRC press, 2013.
%H A371217 Alois P. Heinz, <a href="/A371217/b371217.txt">Table of n, a(n) for n = 1..450</a>
%H A371217 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. 10.
%F A371217 a(n) = 1 + (n-1)*(1 + 2*Sum_{i=1..n-1} (i-1)!*binomial(n-1, i)).
%F A371217 a(n) mod 2 = n mod 2 = A000035(n). - _Alois P. Heinz_, Mar 22 2024
%F A371217 a(n) ~ 2*exp(1)*(n-1)!. - _Vaclav Kotesovec_, Jul 27 2024
%e A371217 Suppose the deck consists of 4 cards (1,2,3,4), and the assistant gets two cards. If the two cards contain 4, the assistant hides 4 and signals it with the other card face down. If there is no 4, then the cards are a and a+1 modulo 3. The assistant hides a+1, and signals it with a.
%p A371217 a:= proc(n) option remember; `if`(n<4, n*(n^2-2*n+2),
%p A371217 ((11*n^2-66*n-61)*a(n-1) -(17*n^2-155*n+134)*a(n-2)
%p A371217 +(n-3)*(n-81)*a(n-3) +(n-4)*(5*n+26)*a(n-4))/(11*n-72))
%p A371217 end:
%p A371217 seq(a(n), n=1..23); # _Alois P. Heinz_, Mar 18 2024
%t A371217 Table[1 + (k - 1)(2 Sum[Binomial[k - 1, i] (i - 1)!, {i, 1, k - 1}] + 1), {k, 20}]
%o A371217 (Python)
%o A371217 from math import factorial
%o A371217 def A371217(n): return n+((n-1)*sum(factorial(n-1)//((i+1)*factorial(n-i-2)) for i in range(n-1))<<1) # _Chai Wah Wu_, May 02 2024
%Y A371217 Cf. A000035, A030495, A275929, A370888.
%K A371217 nonn
%O A371217 1,2
%A A371217 _Tanya Khovanova_ and PRIMES STEP junior group, Mar 15 2024