

A343273


a(n) is the number of geometrically distinct edgeunfoldings of the regular ngonal cupola.


0



308, 3030, 29757, 294327, 2911142, 28814940, 285214743, 2823311133, 27947663768, 276653115090, 2738581182417, 27109156615827, 268352962161482, 2656420444277880, 26295851254778283, 260302091898387033, 2576725065493516028, 25506948561006315150
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OFFSET

3,1


COMMENTS

The term "regular" applies only to the regular ngon and 2ngon (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 twosided (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

Table of n, a(n) for n=3..20.
Zsolt Lengvárszky and Rick Mabry, Enumerating nets of prismlike polyhedra, Acta Sci. Math. (Szeged) 83 (2017), no. 34, 377392.
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(m2)  c(m4). 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*n1)  c(2*n2)  5)/8 + (3 + (1)^n)*c(n)/4.
a(n) = (c(2*n+1) + 5*c(2*n)  c(2*n1)  c(2*n2)  5)/8 + (3 + (1)^n)*c(n)/4 for n >= 3 where c(m) = 10*c(m2)  c(m4) 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, 2n1}]+(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 A159004
Adjacent sequences: A343270 A343271 A343272 * A343274 A343275 A343276


KEYWORD

nonn,easy


AUTHOR

Rick Mabry, Apr 10 2021


STATUS

approved



