A396906 a(n) = number of face diagonals in Johnson solid J_n.

2, 5, 15, 30, 50, 65, 6, 10, 15, 2, 5, 0, 0, 6, 8, 10, 0, 27, 46, 70, 85, 15, 30, 50, 65, 8, 12, 20, 20, 30, 30, 45, 45, 60, 24, 24, 36, 50, 50, 65, 65, 80, 80, 12, 20, 30, 45, 60, 4, 2, 0, 18, 16, 28, 26, 26, 24, 55, 50, 50, 45, 10, 15, 15, 33, 110, 100, 400, 380, 380

Offset

1,1

Links

Paolo Xausa, Table of n, a(n) for n = 1..92 (the full sequence).

Wikipedia, List of Johnson solids.

Formula

a(n) = A396904(n) - A396905(n).

a(n) = (1/2)*(Sum_{k=3..A394913(n)} A394912(n,k)*k*(k-3)).

Example

a(6) = 65 because Johnson solid J_6 (pentagonal rotunda) has 10 triangular faces (no diagonals), 6 pentagonal faces (5 diagonals each) and 1 10-gonal face (35 diagonals), for a total of 10*0 + 6*5 + 1*35 = 65 face diagonals.

Mathematica

Sum[k*(k-3), {k, Map[Length, #]}]/2 & @@@ Array[PolyhedronData[{"Johnson", #}, {"FaceIndices"}] &, 92]

Crossrefs

Cf. A394912, A394913, A396904, A396905.

Sequence in context: A287843 A290828 A290836 * A191313 A078528 A309272

Adjacent sequences: A396903 A396904 A396905 * A396910 A396911 A396912

Keyword

nonn,fini,full

Author

Paolo Xausa, Jun 09 2026

Status

approved