OFFSET

3,1

COMMENTS

The term "regular" applies only to the regular n-gon and 2n-gon (the "top and bottom" of the cupola), the other faces (the "sides") being n isosceles triangles and n sufficiently long rectangles. For n=3,4,5, regular triangles and squares can be used for the sides. That applies to n=6 if a two-sided (flat) polyhedron is allowed.

The first 25 terms of the auxiliary sequence c(n) in the Formula and Mathematica program match the 25 terms listed for sequence A085376.

LINKS

Zsolt LengvĂˇrszky and Rick Mabry, Enumerating nets of prism-like polyhedra, Acta Sci. Math. (Szeged) 83 (2017), no. 3-4, 377-392.

Wikipedia, Cupola

Index entries for linear recurrences with constant coefficients, signature (11,-1,-109,109,1,-11,1).

FORMULA

Recursively define the sequence c(m) as follows: Let c(1) = 1, c(2) = 3, c(3) = 11, c(4) = 30, and for m > 4, let c(m) = 10*c(m-2) - c(m-4). Then for all n >= 3, the sequence a(n) can be given by a(n) = (c(2*n+1) + 5*c(2*n) - c(2*n-1) - c(2*n-2) - 5)/8 + (3 + (-1)^n)*c(n)/4.

a(n) = (c(2*n+1) + 5*c(2*n) - c(2*n-1) - c(2*n-2) - 5)/8 + (3 + (-1)^n)*c(n)/4 for n >= 3 where c(m) = 10*c(m-2) - c(m-4) for m > 4 and c(1) = 1, c(2) = 3, c(3) = 11, c(4) = 30.

G.f.: x^3*(308 - 358*x - 3265*x^2 + 3602*x^3 - 360*x^5 + 33 x^6)/(1 - 11*x + x^2 + 109*x^3 - 109*x^4 - x^5 + 11*x^6 - x^7). - Stefano Spezia, Apr 10 2021

MATHEMATICA

a[n_]:=Sum[c[k], {k, 1, 2n-1}]+(1/2)c[2n]+If[OddQ[n], (1/2)c[n], c[n]];

c[1] = 1; c[2] = 3; c[3] = 11; c[4] = 30;

c[m_] := c[m] = 10 c[m - 2] - c[m - 4];

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Rick Mabry, Apr 10 2021

STATUS

approved