A383974 -id:A383974 - cp's OEIS frontend

A383490 Number of polyforms with n cells on the faces of a rhombic triacontahedron up to rotation and reflection.

1, 1, 1, 3, 4, 12, 24, 65, 156, 422, 1057, 2708, 6574, 15399, 33438, 66787, 118242, 180602, 227896, 226652, 169294, 97224, 44324, 16416, 5005, 1247, 261, 46, 8, 1, 1

Offset: 0

Peter Kagey, Apr 28 2025

These are "free" polyforms.
The rhombic triacontahedron is the polyhedral dual of the icosidodecahedron.
Also the number of free polyedges of the dodecahedron.

Wikipedia, Rhombic triacontahedron

Cf. A383491 (one-sided).
Cf. A030135 (dodecahedron), A030136 (icosahedron), A340635 (deltoidal hexecontahedron), A383492 (triakis icosahedron), A383494 (pentakis dodecahedron), A383496 (disdyakis triacontahedron).
Cf. A383974.

a(14)-a(30) from Pontus von Brömssen, May 23 2025

A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

Original entry on oeis.org

1, 1, 3, 5, 16, 39, 127, 357, 1067, 2861, 7071, 14827, 25638, 33730, 33189, 24838, 14954, 7188, 2905, 912, 254, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 48 symmetries of the rhombic dodecahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 74, 269, 876, 2788, 7639, 17828, 32326, 44375, 46456, 39213, 26865, 15470, 7278, 2917, 913, 254, 49, 11, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 48 symmetries of the cuboctahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 84, 334, 1330, 5495, 22776, 94920, 394706

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the rhombic triacontahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

Original entry on oeis.org

1, 1, 3, 7, 24, 81, 323, 1265, 5202, 21335, 88412, 364897

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks," "polyedges," and "polyforms."

These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other using the 120 symmetries of the icosidodecahedron.

Cf. A019988.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron), A383974 (tetrahedron, row 3), A383981 (rhombic dodecahedron), A383982 (cuboctahedron), A383983 (rhombic triacontahedron), A383984 (icosidodecahedron).

A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 3, 5, 6, 6, 4, 2, 1, 1, 1, 1, 1, 3, 5, 12, 19, 23, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 1, 3, 5, 12, 30, 56, 91, 128, 147, 147, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 1, 3, 5, 12, 30, 79, 180, 364, 633, 961, 1300, 1551, 1644, 1556, 1311, 980, 663, 402, 221, 115, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Peter Kagey, May 16 2025

Connected subsets of edges are also called "polysticks", "polyedges", and "polyforms".
These are "free" polyforms, in that two polyforms are equivalent if one can be mapped to the other via the n! symmetries of the n-simplex.
Equivalently, T(n,k) is the number of connected unlabeled graphs with k edges and between 1 and n+1 vertices. - Pontus von Brömssen, May 27 2025

			Triangle begins:
 0 | 1;
 1 | 1, 1;
 2 | 1, 1, 1, 1;
 3 | 1, 1, 1, 3, 2, 1, 1;
 4 | 1, 1, 1, 3, 5, 6, 6, 4, 2, 1, 1;
 5 | 1, 1, 1, 3, 5, 12, 19, 23, 24, 21, 15, 9, 5, 2, 1, 1;
 6 | 1, 1, 1, 3, 5, 12, 30, 56, 91, 128, 147, 147, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1;

Peter Kagey, Illustration of row 3.

Cf. A333333 (cube, row 3), A383490 (dodecahedron), A383973 (octahedron, row 3), A383974 (icosahedron).

Cf. A000217, A002905, A292300.

T(n,n) = A002905(n).

The sum of row n is A292300(n+1)+1 for n >= 1. - Pontus von Brömssen, May 26 2025

Missing term a(62)=1 inserted and a(73)-a(91) added by Pontus von Brömssen, May 26 2025

cp's OEIS Frontend

A383490 Number of polyforms with n cells on the faces of a rhombic triacontahedron up to rotation and reflection.

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A383981 Number of connected subsets of n edges of the rhombic dodecahedron up to the 48 rotations and reflections of the rhombic dodecahedron.

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A383982 Number of connected subsets of n edges of the cuboctahedron up to the 48 rotations and reflections of the cuboctahedron.

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A383983 Number of connected subsets of n edges of the rhombic triacontahedron up to the 120 rotations and reflections of the rhombic triacontahedron.

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A383984 Number of connected subsets of n edges of the icosidodecahedron up to the 120 rotations and reflections of the icosidodecahedron.

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A383975 Irregular triangle: T(n,k) gives the number of connected subsets of k edges of the n-simplex up to isometries of the n-simplex, with 0 <= k <= A000217(n).

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