cp's OEIS Frontend

A(0,0) = 2 by the convention that 0^0 = 1, in spirit with A003992.

A(n,2) = A000616(n), because the Boolean functions' truth tables are n-dimensional edge-length-2 hypercubes, and are considered equivalent under action of input permutation (transposition of dimensions) and applications of NOT to inputs (reflection in dimensions) that compose for the hypercube's symmetry group.

A(n,3) is the number of transitions in an isotropic non-totalistic Life-like cellular automaton with an n-dimensional Moore neighborhood.

A(n,k) ~ 2^(k^n)/(n!*2^n) for large k where n >= 1, and large n where k >= 2, and converges from above. Proof: When computing it with the Pólya enumeration theorem (for each action, count colorings of cycles under it instead of cells, average result over actions), the asymptotic form describes the number of states contributed by the identity action. Over k, the values contributed by the other actions are at most O(2^(k^n/2)), so the proportion that they contribute may be made arbitrarily small by choosing a large enough n. Over n, there are no more than n!*2^n such non-identity actions, assuming that they are all of order 2 (as an upper bound). Each one may have no more than a hyperplane (2^k^(n-1)) of fixed points, which (if it is of order 2) will multiply its colorings by 2^(k^(n-1)/2). The ratio of the identity term to the others is at least O(2^k^n/(n!*2^n*2^(k^(n-1)/2))), and its base-2 logarithm, by Stirling's approximation, is O(k^n-n*log(n)-n-k^(n-1)/2), so the 2^(k^n) term will dominate.

A(1,k) is the only row >= 1 that is linear-recurrent over k (and has a rational generating function), all other nontrivial rows and columns grow faster than any linear-recurrent function.

A000120(A(n,k)) is eventually periodic over k if and only if n <= 2.

DTM (distance to mate) is measured in ply (half-moves), equivalence is under action of the square's symmetry group (D_4), however colors can't be swapped because one side always has a rook.

A225552(n) is the maximum value of k such that A(n,k)!=0, divided by 2 because it's in moves instead of ply (though each column seems to terminate at an even (losing) k, so perhaps no information is lost).

In each column, there are n-2 positions counted (in which the king without the rook is to move) that are orphans (with no predecessors), each represented by a horizontally-shifted version of the piece configuration (Ka1 - Kc2 Rc1) (in which the rook is blocked from having been moved downwards (to cause the check) by its king, that could not have exited from in front without having been adjacent to the king to move). Each one's DTM corresponds with the distance of the king to move from the corner in the opposite direction from the other king, DTM is 4 when on a1, 12 when on b1, 18 when on c1, 22 when on d1, 28 when on e1, and increasing by 2 for all thereafter.

Positions with the side with the king to move at a1, its rook at b1 and the opponent king at (ceiling(n/2),ceiling(n/2)) seem to be wins in 4*n - 6 + 2*floor((n+1)/2) - 2*floor(n/6).

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For fixed values of n, all values A(n,k) with odd k (winning positions) seem to exceed A(n,k-1) and A(n,k+1) (losing positions) until a threshold value, beyond which this relationship inverts.

For all k, A(n,k)/Sum_{k>=0}(A(n,k)) is o(1) over n (for n>=A225552^(-1)(2*k)), so there is no k such that the polynomial eventually describing A(n,k) is of degree 6, and for all j, Sum_{k=0..j}(A(n,k)) is o(Sum_{k>=0}(A(n,k))).

For each k, there exists a polynomial eventually describing A(n,k) over n, though there can be arbitrarily many leading zero terms so it can take arbitrarily long to do so. Note that it is not necessarily an upper bound.

All polynomials for k >= 7 are quadratic.

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Conjecture 1: If A(n+1,k) is nonzero, it is always greater than A(n,k).

Conjecture 2: As the increments to A225552 become periodic for n>24, so too do the differences between successive polynomials over n for each k>108, and for all j, Sum_{k=2*A225552(n)-j..2*A225552(n)}(A(n,k)) is o(Sum_{k>=0}(A(n,k))) (the same equation from the opposite side).

Conjecture 3: (Start)

Note that Sum_{k>=0}(A(n,k)-A(n-1,k)) ~ 3*n^5/2. (See FORMULA for the exact form.)

For all n, assuming conjecture 2, because the polynomials provide upper bounds and all are of degree <=5, it is necessary for the sum of all nonzero A(n,k)'s polynomials (for k<=2*A225552(n)) to have n^2 coefficients summing to 3*n^4/2.

A225552(n) ~ 7*n/3 (and seems to become periodic for n>24), so the number of elements added with each increment of n is ~ 14/3.

Thus, for n>24, the change of 3/2 to the sum of n^5 coefficients with each increment to n causes each new term's eventual polynomial to have an n^2 term with a coefficient asymptotic to (3/2)/(14/3)*n^2=9*n^2/28. (End)

Conjecture 4: The function o (defined in the FORMULA section), for each parity of k, contains only polynomials over n (without any terms with coefficients of n%2 or (-1)**n). For all m>=3, there exists a pair of polynomials over n (for even and odd k), such that when o adds them to the output of the k-specific fitting polynomial for n=k-m, the values returned are correct for all k>=5*(m-1). (Works for m=2,3,4.) Equivalently, for all k>10, while A(n,k) can be described exactly by a k-specific polynomial over n for all n>=A052928(k), it can be described exactly by the sum of that and a (2*k-n)-specific polynomial over n (or equivalently, over k) for all n>=k-1-floor(k/5).

The number of king positions over which you iterate when making tablebases of positions containing pawns, wherein it is only equivalent under reflection in the x axis.

Rotations and reflections of placements are not counted. (If they were then see A035286.)

a(8)=462 is the number of states in the KvK endgame in an eightfold-reducing chess tablebase on 8 X 8 boards.

When kings are unlabeled, see A279111. The ratio a(n)/A279111(n) is bounded in the interval [1, 2] and converges to 2, because the number of placements in which the kings' positions can be swapped by an automorphism is O(n^2), while the sequence itself is O(n^4).

When there are pawns on the board and the position is only equivalent under reflection in the x axis, see A357740.

A quasipolynomial of degree 4 and period 2. - Charles R Greathouse IV, Feb 02 2023