A337191 If cards numbered 1 through n are "Down Two Table" shuffled (top two put on bottom one at a time, third from top card dealt to table) until all of the cards are placed on the table, a(n) is the number of the last card dealt.

1, 1, 1, 4, 4, 1, 7, 4, 1, 7, 4, 10, 7, 13, 10, 16, 13, 1, 16, 4, 19, 7, 22, 10, 25, 13, 1, 16, 4, 19, 7, 22, 10, 25, 13, 28, 16, 31, 19, 34, 22, 37, 25, 40, 28, 43, 31, 46, 34, 49, 37, 52, 40, 1, 43, 4, 46, 7, 49, 10, 52, 13, 55, 16, 58, 19, 61, 22, 64, 25, 67

Offset

1,4

Comments

This sequence is related to the Josephus Problem, which can be modeled with the Australian Under Down Shuffle, one card placed under the deck, one card laid down on the table until all the cards are on the table.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for sequences related to the Josephus Problem

Formula

a(1) = 1, a(2) = 1, a(n) = (a(n-2) + 3) (mod n) if (a(n-2) + 3) (mod n) is not 0; a(n) = n if (a(n-2) + 3) (mod n)=0.

Any number n can be written as either 2*(3^k) + 2m (where 0 <= m < 3^k, k = 0,1,2,...) or 3^k + 2m (where 0 <= m < 3^k, k = 0,1,2,...), in either case a(n) = 3m + 1.

Mathematica

nxt[{n_, a_, b_}]:={n+1, b, If[Mod[a+3, n+1]!=0, Mod[a+3, n+1], n+1]}; NestList[nxt, {2, 1, 1}, 70][[;; , 2]] (* Harvey P. Dale, Jul 27 2024 *)

Prog

(PARI) a(n) = if (n <= 2, 1, my(x = (a(n-2) + 3) % n); if (x, x, n)); \\ Michel Marcus, Aug 20 2020

Crossrefs

Cf. A006257.

Sequence in context: A247252 A016495 A335826 * A341863 A047213 A128213

Adjacent sequences: A337188 A337189 A337190 * A337192 A337193 A337194

Keyword

nonn

Author

Robert W. Vallin, Aug 18 2020

Extensions

More terms from Michel Marcus, Aug 20 2020

Status

approved