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A343273
a(n) is the number of geometrically distinct edge-unfoldings of the regular n-gonal cupola.
0
308, 3030, 29757, 294327, 2911142, 28814940, 285214743, 2823311133, 27947663768, 276653115090, 2738581182417, 27109156615827, 268352962161482, 2656420444277880, 26295851254778283, 260302091898387033, 2576725065493516028, 25506948561006315150
OFFSET
3,1
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.
Zsolt LengvĂˇrszky and Rick Mabry, Enumerating nets of prism-like polyhedra, Acta Sci. Math. (Szeged) 83 (2017), no. 3-4, 377-392.
Wikipedia, Cupola
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
Cf. A085376; see the sequence c(n) in the Formula and Mathematica program, but note that A085376 has only been conjectured to be the same as c(n).
Sequence in context: A237457 A234211 A053172 * A337955 A091552 A364936
KEYWORD
nonn,easy
AUTHOR
Rick Mabry, Apr 10 2021
STATUS
approved

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Last modified September 20 03:21 EDT 2024. Contains 376016 sequences. (Running on oeis4.)