A060852 -id:A060852 - cp's OEIS frontend

A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

4, 8, 6, 20, 12

Offset: 1

Henry Bottomley, Aug 14 2001

The preferred order for these five numbers is 4, 6, 8, 12, 20 (tetrahedron, octahedron, cube, icosahedron, dodecahedron), as in A053016. - N. J. A. Sloane, Nov 05 2020
Also number of faces of Platonic solids ordered by increasing ratios of volumes to their respective circumscribed spheres. See cross-references for actual ratios. - Rick L. Shepherd, Oct 04 2009
Also the expected lengths of nontrivial random walks along the edges of a Platonic solid from one vertex back to itself. - Jens Voß, Jan 02 2014

			a(2) = 8 since a cube has eight vertices.

Cf. A053012, A053016, A060296, A060852.
Cf. A165922 (tetrahedron), A049541 (octahedron), A165952 (cube), A165954 (icosahedron), A165953 (dodecahedron). - Rick L. Shepherd, Oct 04 2009
Cf. A234974. - Jens Voß, Jan 02 2014

a(n) = A063722(n) - A053016(n) + 2.

A063722 Number of edges in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

Original entry on oeis.org

6, 12, 12, 30, 30

Offset: 1

Henry Bottomley, Aug 14 2001

			a(2) = 12 since a cube has twelve edges.

Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A053012, A060296, A060852.

a(n) = A053016(n)+A063723(n)-2.

A137457 Consider a row of standard dice as a counter. This sequence enumerates the number of changes (one face rotated over an edge to an adjacent face) from n-1 to n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1

Offset: 0

Robert G. Wilson v, Apr 18 2008

nonn

Most counters 'zero' out at '0' but the dice 'zero' out at '1' which is the initial state. So to increment 1 -> 2 requires 1 move, 2 -> 3 requires 1 move, 3 -> 4 requires 2 moves, 4 -> 5 requires 1 move, 5 -> 6 requires 1 move and 6 -> 0 requires 2 moves.

First occurrence of k (A026532): 1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, ....

Eric Weisstein's World of Mathematics, Dice.
Wikipedia, Dice.

Cf. A057436, A060852, A026532, A133794.

f[n_] := Block[{a = IntegerDigits[n - 1, 6] + 1, b = IntegerDigits[n, 6] + 1, c}, If[Length@b > Length@a, a = Prepend[a, 1]]; c = Transpose[{a, b}] /. {{d_, d_} -> 0, {1, 2} -> 1, {2, 3} -> 1, {3, 4} -> 2, {4, 5} -> 1, {5, 6} -> 1, {6, 1} -> 2}; Plus @@ c]; Array[f, 105]

cp's OEIS Frontend

A063723 Number of vertices in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A063722 Number of edges in the Platonic solids (in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron).

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A137457 Consider a row of standard dice as a counter. This sequence enumerates the number of changes (one face rotated over an edge to an adjacent face) from n-1 to n.

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Mathematica