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These are unoriented polyominoes of the regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one. The dimension of the convex hull of the cell centers determines the dimension d. - Robert A. Russell, Aug 09 2022

T(n,k) is also the number of n-celled polyominoes made up of k-dimensional cubes, counted up to rotation, reflection, and translation.

An orthoplex polyomino does not extend more than two units along any axis.

The code used here is similar to the binary code defined in A365139, but it is based on an ordering of all sequences of nonnegative integers with a finite number of nonzero terms. The ordering is defined by assigning the ordinal number (Product_{i>=1} prime(i)^x_i) - 1 to the sequence (x_1, x_2, ...). Given a polyomino in any dimension (represented as a finite set of sequences of nonnegative integers with a finite number of nonzero terms, each representing a cell of the polyomino), consider all the ways of rotating/reflecting it. Translate each such rotation/reflection so that the minimum coordinate is 0 in each dimension, and add the powers of 2 with exponents equal to the ordinal numbers of the cells of the translation. The code of the polyomino is the minimum of those sums. The minimum will occur for a transformation for which the largest index of a nonzero coordinate is as small as possible, so only a finite number of transformations need to be considered.

Can be read as an irregular triangle, whose n-th row contains A005519(n) terms. The first term of the n-th row is 2^n-1. For n <= 4, the last term of the n-th row corresponds to the straight n-omino (with code Sum_{j=0..n-1} 2^(2^j-1)), but the last term of the 5th row (133158) corresponds to the x-pentomino, and the last term of the 6th row (576460752840441890) corresponds to the hexacube composed of two straight tricubes orthogonally attached to each other at their middle cubes.

Can be read as an irregular triangle, whose n-th row contains A005519(n) terms. The first term of the n-th row is A000720(n). The number of times d occurs in the n-th row is A049430(n,d).

a(1/2 * n * (n-1) + d) gives values of Shapes(n, d) for n > 0, d > 0, d <= n. For d > n, use a(1/2 * n * (n + 1)) or A005519's a(n).

A polytopomino shape is a shape constructed of n contiguous hypercubes that is invariant over rotation but not necessarily over 'flipping', i.e. mirror images are distinct. See example for details of when flipping is or is not considered.

A000988 gives values for Shape(n, 2), e.g. a(1/2 * n * (n - 1) + 2) and A000162 for Shape(n, 3), e.g. a(1/2 * n * (n - 1) + 3.

A005519 gives values for Shape(n, n), e.g. a(1/2 * n * (n + 1)). These shapes always can be expressed using only n-1 dimensions and therefore contain no mirror-image or "flipped" shapes.

Shape(n, d) is the union of all shapes with n points that can be expressed in a dimension x for x from 1 to d - 1 where "flipped" shapes are excluded plus all shapes with n points that must be expressed by at least d dimensions where "flipped" shapes are included. A049429 gives values for x in [1, d - 1] and n.

Specificly, if b(m) defines the sequence A049429, b(1/2 * n * (n + 1) + x) is the term for all shapes that must be expressed in at least x dimensions and containing n points.

There is no sequence that describes shapes with n points and exactly d dimensions for which "flipped" shapes are considered distinct, so this formula cannot be completely expressed as the sum of other formula.

The main difficulty in computing this sequence is in a) the fast implemention of a set (as in a set of points [shape] and a set of shapes [Shape(n, d)]), especially with respect to rotation of points and b) the difficulty in eliminating duplicate entries.

The later case is difficult because in order to determine whether two shapes are the same, one must compute all possible R in order to determine the R that may orient shape X the same as shape Y. The translation vector T is uniquely given based on R but requires finding the minimum point of the bounding hypercube of each shape that is linear with respect to d.

Ideally, a good algorithm for b must be found, especially if a "definitive orientation" can be determined such that all shapes will be oriented using the definitive orientation before being compared and thus the comparison consists only of comparing the points in X and Y to make sure they are the same.

Also, it should be possible to reduce the loop over Cardinal Vectors since some vectors are equivalent, such as adding (1, 0) or (-1, 0) to the point (0, 0) since the shape has symmetry and therefore both new shapes are equivalent.