login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=3..20.

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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 21:57 EST 2022. Contains 358453 sequences. (Running on oeis4.)