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 A349697 #18 Dec 15 2021 02:13:01 %S A349697 78307,186990749618019112, %T A349697 1614205257536455860879998130775735700828260230275, %U A349697 10794889897425456513785608689552167481027910004023676512263195628077085109766959836330846217 %N A349697 Numerators of the probability that the first player wins the game Super Six if both players have n sticks in their hand and if there are 3 sticks on the lid, assuming optimal play. %C A349697 The rules of Super Six for two players are as follows. The equipment consists of a six-sided die, a number of sticks, and a box whose lid has six holes. The holes numbered 1 through 5 are shallow, and a stick placed in any one of them will stand up in it; hole #6 goes all the way through the lid so that any stick placed in it falls into the box and is out of play. Initially, an even number of sticks are divided evenly between the two players. The goal is to get rid of all one's sticks before the other player does. %C A349697 The players take turns. On each turn, the active player rolls the die and places a stick in the numbered hole that matches the number on the die (e.g., a player who rolls a 4 then places a stick in hole #4). The player may roll and place a stick for each roll as many times as desired until rolling a number that is already filled by a stick. When this occurs, the player must take that stick in hand, and play passes to the opponent. %C A349697 The game proceeds with players taking turns and ends when one player has run out of sticks. The only freedom that the players have is the decision whether to continue rolling the die or not after successfully placing a stick. %C A349697 The optimum strategy and the winning probabilities can be found in "Optimum Strategies for the Game Super Six" (see link below). The terms of this sequence give the numerators and the terms of sequence A349698 give the denominators of the probability that the first player wins if there are 3 sticks on the lid and both players hold n sticks in their hands, assuming optimal play. If n tends to infinity this probability tends to 1/2. %H A349697 Ruediger Jehn, <a href="/A349697/b349697.txt">Table of n, a(n) for n = 1..11</a> %H A349697 Rüdiger Jehn, <a href="https://arxiv.org/abs/2109.10700">Optimum Strategies for the Game Super Six</a>, arXiv:2109.10700 [math.GM], 2021. %H A349697 Wikipedia, <a href="https://de.wikipedia.org/wiki/Super_Six_(Spiel)">Super Six</a> (in German) %e A349697 a(1) = 78307 because the probability that the first player wins the game Super Six, when both players have 1 stick and there are 3 sticks on the lid, is 78307/127838 (0.612548...). %Y A349697 Cf. A345383, A349698. %K A349697 nonn,frac %O A349697 1,1 %A A349697 _Ruediger Jehn_, _Kester Habermann_ and _Pontus von Brömssen_, Nov 25 2021