A296602 Values of F for which there is a unique convex polyhedron with F faces that are all regular polygons.

4, 19, 23, 25, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173

Offset

1,1

Comments

The main entry for this sequence is A180916.

All terms except 4 are odd, because both the cube and the pentagonal pyramid have 6 faces, and for any even F > 6 both a prism and an antiprism can have F faces. Platonic solids, Archimedean solids, Johnson solids, and prisms account for the missing odd numbers.

Links

Table of n, a(n) for n=1..75.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

A180916(a(n)) = 1.

From Colin Barker, Jul 05 2020: (Start)

G.f.: x*(4 + 11*x - 11*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^8 - 2*x^9 + 2*x^12 - 2*x^13) / (1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>14.

(End)

Example

The regular tetrahedron is the only convex polyhedron with 4 faces that are all regular polygons, and no such polyhedron with fewer than 4 faces exists, so a(1) = 4.

Mathematica

LinearRecurrence[{2, -1}, {4, 19, 23, 25, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51}, 30] (* Georg Fischer, Oct 26 2020 *)

Crossrefs

Cf. A180916, A242731, A296603, A296604.

Sequence in context: A012879 A072178 A116980 * A022135 A192193 A028564

Adjacent sequences: A296599 A296600 A296601 * A296603 A296604 A296605

Keyword

nonn,easy

Author

Jonathan Sondow, Jan 28 2018

Status

approved