A222192 -id:A222192 - cp's OEIS frontend

A333333 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to isometries of the n-cube, with 0 <= k <= A001787(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 9, 14, 19, 16, 9, 4, 1, 1, 1, 1, 1, 3, 7, 21, 72, 269, 994, 3615, 12337, 38603, 107720, 259990, 526314, 865217, 1139344, 1225762, 1109138, 857376, 574284, 333484, 169023, 73994, 28222, 9138, 2595, 604, 140, 24, 6, 1, 1

Offset: 1

Peter Kagey, Mar 15 2020

This sequence counts edge-induced connected subgraphs of the n-dimensional hypercube graph, up to automorphisms of the hypercube; A369605 counts vertex-induced such graphs. - Pontus von Brömssen, May 12 2025

Row 3 gives the number of polyforms with n cells on the faces of a rhombic dodecahedron up to rotation and reflection. - Peter Kagey, May 19 2025

			Table begins:
n\k| 0  1  2  3  4   5   6    7    8     9     10     11      12 ...
---+----------------------------------------------------------------
  1| 1, 1;
  2| 1, 1, 1, 1, 1;
  3| 1, 1, 1, 3, 4,  9, 14,  19,  16,    9,     4,     1,      1;
  4| 1, 1, 1, 3, 7, 21, 72, 269, 994, 3615, 12337, 38603, 107720, ...

Code Golf Stack Exchange, Counting polyominoes on (hyper-)cubes
Peter Kagey, Examples of T(3,3) through T(3,8).
Wikipedia, Hypercube
Wikipedia, Polystick

Cf. A001787, A222192, A369605.

T(n,k) = T(n-1,k) for k < n.
T(n,0) = T(n,1) = T(n,2) = T(n,A001787(n)-1) = T(n,A001787(n)) = 1.
A222192(n) = Sum_{k=0..n*2^(n-1)} T(n,k) - Sum_{k=0..(n-1)*2^(n-2)} T(n-1,k) for n >= 2. - Peter Kagey, Jun 19 2023

a(31)-a(40) from Pontus von Brömssen, May 12 2025

a(41)-a(53) from Pontus von Brömssen, May 30 2025

A222186 a(n) = number of distinct ways to choose a subset of the n*2^(n-1) edges of the n-cube so that the resulting figure is connected and fully n-dimensional.

Original entry on oeis.org

1, 3, 123, 14632581

Offset: 1

N. J. A. Sloane, Feb 11 2013

"Distinct" means that figures differing by a rotation are not regarded as different.
"Fully n-dimensional" means not lying in a proper subspace.
Suggested by Sol LeWitt's work "Variations of Incomplete Open Cubes," which shows 122 of the 123 figures in the three-dimensional case.

			For n=2 the three figures are: the four edges of a square, or omit one edge, or omit two adjacent edges.

Peter Schjeldahl, Less is beautiful, The Art World, The New Yorker, March 13, 2000, pp. 98-99.

Sol LeWitt, Variations of Incomplete Open Cubes [The full cube itself is not included in his list.]
Andrew Weimholt, 3D solutions in numerical representation
Andrew Weimholt, Notes on reading the 3D solutions
Andrew Weimholt, C++ program for A222186 and A222192

Cf. A222192.

a(3) confirmed by Andrew Weimholt, Feb 12 2013

a(4) computed by Andrew Weimholt, Feb 13 2013

cp's OEIS Frontend

A333333 Irregular triangle: T(n,k) gives the number of k-polysticks on edges of the n-cube up to isometries of the n-cube, with 0 <= k <= A001787(n).

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