|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|