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

Author

Jonathan Sondow, Jan 28 2018

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A180916, A242731, A296603, A296604.

Programs

Mathematica

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

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)