cp's OEIS Frontend

The Scheme-program given cannot in practice compute this further than n=21, as A106485(A126011(22))=36893488147419103232. However, the further terms could be deduced by other means. This sequence is a permutation of the nonnegative integers because combinatorial games form a group under (game) addition and each game has a well-defined, unique negative.

Consider the following rooted trees useful in the combinatorial game theory: each tree has zero or more subtrees at its left side and zero or more subtrees at its right side. The orientation of the subtrees among the other branches of the same side is not distinguished and all the subtrees of the same side are distinct from each other. These kinds of trees map bijectively to nonnegative integers by the following map f: f(empty tree) = 0 and f(tree with left subtrees Tl1, ..., Tlj and right subtrees Tr1, ..., Trk) = 2^(2*f(Tl1)) + ... + 2^(2*f(Tlj)) + 2^((2*f(Tr1))+1) + ... + 2^((2*f(Trk))+1). The ten game trees given on page 40 of "Winning Ways" thus translate to values Game 0 -> 0, Game 1 -> 1, Game -1 -> 2, Game 2 -> 4, Game 1/2 -> 9, Game 1/4 -> 524289, Game * -> 3, Game 1* -> 12, Game *2 -> 195, Game ^ (up) -> 129. However, this correspondence is not bijective with the computed equivalence classes of games, as many integers map to game trees with dominated or reversible options. Here a(n) gives the total number of edges in the tree f(n).

Here we use the encoding explained in A106486. A(i,j) = A(A106485(j),A106485(i)).