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 Lengvárszky 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