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 A374810 #33 Aug 06 2024 22:45:12
%S A374810 1,6,7,12,13,18,23,24,38,39,44,45,50,51,56,62,77,115,121,153,312,333,
%T A374810 350,427,553,554,579
%N A374810 Values k such that the two-player impartial {0,1}-Toggle game on a path P(k+2) = v(1)v(2)...v(k+2) with a (1^k,0,1)-weight assignment is a second-player winning game.
%C A374810 The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.
%C A374810 A Toggle move consists of selecting a vertex v and switching its weight as well as the weight of each of its neighbors. This move is legal only provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.
%C A374810 In the special case G = P(k+2), a (1^k, 0, 1)-weight assignment is one in which vertex v(k+1) is assigned weight 0 and all remaining vertices are assigned weight 1.
%D A374810 E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.
%H A374810 K. Barker, M. DeStefano, E. Fiorini, M. Gohn, J. Miller, J. Roeder, and T. W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Wong/wong43.html">Generalized Impartial Two-player Pebbling Games on K3 and C4</a>, Journal of Integer Sequences, 27(5), 2024.
%H A374810 Matthew Cohen, <a href="/A374810/a374810_2.py.txt">Python</a>
%H A374810 E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Wong/wong31.pdf">Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs</a>, Journal of Integer Sequences, 24(6), 2021.
%e A374810 For n = 6, the {0,1}-Toggle game on P(8) with a (1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
%e A374810 For n = 12, the {0,1}-Toggle game on P(14) with a (1,1,1,1,1,1,1,1,1,1,1,1,0,1)-weight assignment is a second-player winning game.
%o A374810 (Python) # See Cohen link.
%Y A374810 Cf. A071426, A340631, A346197, A346401, A346637, A361517, A374910, A374920
%K A374810 nonn,more
%O A374810 1,2
%A A374810 _Patrick G. Cesarz_, _Matthew Cohen_, _Eugene Fiorini_, _Joshua Lowrance_, _Emily Riley_, _Angel Pedro Torres_ and _Andrew Woldar_, Jul 20 2024