%I #44 Nov 12 2018 01:50:19

%S 2,3,7,11,17,23,27,31,37,41,45,57,61,65,75,79,91,95,99,109,113,125,

%T 129,133,143,147,159,163,167,177,181,193,197,201,211,215,227,231,235,

%U 245,249,261,265,269,279,283,295,299,303,313,317,329,333,337,347,351

%N Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy.

%C The edge-delete game on the graph G is as follows: Two players alternate turns, permanently deleting one edge from G on each turn. The game ends when a vertex is isolated in what remains of G. The player whose deletion created the isolated vertex loses the game.

%C Here P_{n} refers to the path with n vertices, not n edges.

%C a(n)-1 gives the zeros in the nim-sequence of octal game .4, see A002187. - _Zachary Winkeler_, Jul 10 2016

%H Andrew Howroyd, <a href="/A274161/b274161.txt">Table of n, a(n) for n = 1..1000</a>

%H Pratik Alladi, Neel Bhalla, Tanya Khovanova, Nathan Sheffield, Eddie Song, William Sun, Andrew The, Alan Wang, Naor Wiesel, Kevin Zhang Kevin Zhao, <a href="https://arxiv.org/abs/1707.07201">PRIMES STEP Plays Games</a>, arXiv:1707.07201 [math.CO], 2017, Section 8.

%H R. P. Gallant, G. Gunther, B. L. Hartnell, D. F. Rall, <a href="https://www.researchgate.net/publication/268659207">A game of edge removal on graphs</a>, JCMCC, 57 (2006), 75 - 82.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1)

%F The sequence consists of {2,3,17,37} along with all positive integers congruent to 7, 11, 23, 27, and 31 modulo 34.

%F a(n) = a(n-1) + a(n-5) - a(n-6) for n > 15. - _Andrew Howroyd_, Nov 11 2018

%e The number 7 is included because a path on 7 vertices has no winning strategy for player 1 (P1). Consider the edges labeled 1 through 6, left to right along the path. Without loss of generality, P1's first turn is 1, 2, or 3. P1 cannot delete 1 (an immediate loss). If P1 deletes 2, P2 deletes 4 or 5 to force an immediate loss on P1's next turn. If P1 deletes 3, P2 deletes 5 to force the loss.

%t Union[{2, 3, 17, 37}, Flatten[Outer[Plus, {7, 11, 23, 27, 31}, 34 Range[0, 10]]]]

%o (PARI) a(n)=if(n<10, [2, 3, 7, 11, 17, 23, 27, 31, 37][n], [7, 11, 23, 27, 31][n%5+1] + (34*(n\5-1))); \\ _Andrew Howroyd_, Nov 11 2018

%Y Cf. A002187.

%K nonn,easy

%O 1,1

%A _Jennifer A. Moon_ & _Kellen Myers_, Jun 17 2016