%I #28 Oct 29 2020 10:01:15

%S 4,19,23,25,29,31,33,35,39,41,43,45,49,51,53,55,57,59,61,63,65,67,69,

%T 71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,

%U 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

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

%C The main entry for this sequence is A180916.

%C 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.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F A180916(a(n)) = 1.

%F From _Colin Barker_, Jul 05 2020: (Start)

%F 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.

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

%F (End)

%e 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.

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

%Y Cf. A180916, A242731, A296603, A296604.

%K nonn,easy

%O 1,1

%A _Jonathan Sondow_, Jan 28 2018