A296604 Number of Johnson solids with n faces.

0, 0, 0, 0, 1, 2, 1, 4, 2, 4, 3, 4, 2, 8, 1, 3, 3, 4, 0, 6, 1, 4, 0, 2, 0, 4, 3, 0, 0, 1, 0, 5, 0, 1, 0, 0, 3, 0, 0, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,6

Comments

Sum_{n>0} a(n) = 92, the number of Johnson solids, as conjectured by Johnson and proved by Zalgaller.

a(n) > 0 if and only if n is a member of A296603.

Links

Table of n, a(n) for n=1..89.

Norman W. Johnson, Convex Polyhedra with Regular Faces, Canadian Journal of Mathematics, 18 (1966), 169-200.

Eric Weisstein's World of Mathematics, Johnson Solid.

Wikipedia, List of Johnson solids.

Victor A. Zalgaller, Convex Polyhedra with Regular Faces, Zap. Nauchn. Sem. LOMI, 1967, Volume 2. Pages 5-221 (Mi znsl1408).

Index entries for linear recurrences with constant coefficients, signature (1).

Formula

a(62) = 5.

a(n) = 0 for n > 62.

Example

The square pyramid is the only Johnson solid with five faces, so a(5) = 1.

Mathematica

BinCounts[PolyhedronData["Johnson", "FaceCount"], {1, 100}] (* Paolo Xausa, Apr 06 2026 *)

Prog

(Python) A296604 = lambda n: sum(len(p['faces'])==n for p in JS) # see A394913 for definition of JS. - M. F. Hasler, Apr 10 2026

Crossrefs

Cf. A181708, A242731, A296602, A296603.

Sequence in context: A070556 A277687 A065295 * A261211 A233521 A035685

Adjacent sequences: A296601 A296602 A296603 * A296605 A296606 A296607

Keyword

nonn,easy

Author

Jonathan Sondow, Jan 28 2018

Status

approved