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 A371217 The maximum deck size to perform Colm Mulcahy's n-card trick. 6
 1, 4, 15, 52, 197, 896, 4987, 33216, 257161, 2262124, 22241671, 241476060, 2867551117, 36960108680, 513753523571, 7659705147976, 121918431264273, 2063255678027668, 36991535865656959, 700377953116334788, 13963866589144933461, 292421219327021540176, 6417047546280200867819 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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. REFERENCES Wallace Lee, Math Miracles, published by Seeman Printery, Durham, N.C., 1950. Colm Mulcahy, Mathematical card magic: fifty-two new effects, published by CRC press, 2013. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..450 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. 10. FORMULA a(n) = 1 + (n-1)*(1 + 2*Sum_{i=1..n-1} (i-1)!*binomial(n-1, i)). a(n) mod 2 = n mod 2 = A000035(n). - Alois P. Heinz, Mar 22 2024 a(n) ~ 2*exp(1)*(n-1)!. - Vaclav Kotesovec, Jul 27 2024 EXAMPLE 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. MAPLE a:= proc(n) option remember; `if`(n<4, n*(n^2-2*n+2), ((11*n^2-66*n-61)*a(n-1) -(17*n^2-155*n+134)*a(n-2) +(n-3)*(n-81)*a(n-3) +(n-4)*(5*n+26)*a(n-4))/(11*n-72)) end: seq(a(n), n=1..23); # Alois P. Heinz, Mar 18 2024 MATHEMATICA Table[1 + (k - 1)(2 Sum[Binomial[k - 1, i] (i - 1)!, {i, 1, k - 1}] + 1), {k, 20}] PROG (Python) from math import factorial 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 CROSSREFS Cf. A000035, A030495, A275929, A370888. Sequence in context: A329253 A161125 A027295 * A208722 A057332 A230623 Adjacent sequences: A371214 A371215 A371216 * A371218 A371219 A371220 KEYWORD nonn AUTHOR Tanya Khovanova and PRIMES STEP junior group, Mar 15 2024 STATUS approved

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Last modified September 13 04:07 EDT 2024. Contains 375859 sequences. (Running on oeis4.)