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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.
(Formerly M1676)
7
1, 2, 6, 25, 107, 509, 2468, 12258, 61797, 315830, 1630770, 8498303, 44629855, 235974495, 1255105304, 6710883952, 36050676617, 194478962422, 1053120661726, 5722375202661, 31191334491891, 170504130213135, 934495666529380, 5134182220623958, 28270742653671621 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
COMMENTS
Total number of dissections of an n-gon into polygons without reflection and rooted at a cell. - Sean A. Irvine, May 14 2015
Say two n-gons are equivalent (or in the same convexity class) if there is a bijection between the edges and vertices which preserves inclusion of vertices and edges, preserves the handedness of the polygon (does not reflect the polygon over a line), maps vertices of the convex hulls to each other, and induces an equivalence on each nontrivially connected component of Hull(X) \ X. This sequence is the number of convexity classes for an n-gon, up to rotation. - Griffin N. Macris, Mar 02 2021
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
FORMULA
G.f.: -f(x) - (f(x)^2 + f(x^2))/2 + Sum_{k>=1} (phi(k)/k)*log(1/(1 - f(x^k))), where phi(k) is Euler's Totient function and f(x) = (1 + x - sqrt(1 - 6x + x^2))/4 is related to the o.g.f. for A001003. - Griffin N. Macris, Mar 02 2021
PROG
(PARI) \\ See A003442 for DissectionsModCyclicRooted.
DissectionsModCyclicRooted(apply(i->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017
CROSSREFS
Cf. A003442.
Sequence in context: A064811 A074418 A330637 * A199241 A366236 A276277
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 22 2017
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 March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)