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!)
A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.
(Formerly M3330)
4
1, 4, 8, 16, 25, 40, 56, 80, 105, 140, 176, 224, 273, 336, 400, 480, 561, 660, 760, 880, 1001, 1144, 1288, 1456, 1625, 1820, 2016, 2240, 2465, 2720, 2976, 3264, 3553, 3876, 4200, 4560, 4921, 5320, 5720, 6160, 6601, 7084, 7568, 8096, 8625, 9200, 9776, 10400 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,2
COMMENTS
In other words, the number of 2-dissections of an n-gon modulo the cyclic action.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
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.: x^5 * (1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2).
See also the Maple code for an explicit formula.
a(n) = A006584(n+3) - A027656(n). - Yosu Yurramendi, Aug 07 2008
a(n) = (n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24, for n>=5. - Luce ETIENNE, Apr 04 2015
MAPLE
T51:= proc(n)
if n mod 2 = 0 then n*(n-2)*(n-4)/12;
else (n+1)*(n-3)*(n-4)/12; fi end;
[seq(T51(n), n=5..80)]; # N. J. A. Sloane, Dec 28 2012
MATHEMATICA
Table[((n - 4) (2 n^2 - 4 n - 3 (1 - (-1)^n)) / 24), {n, 5, 60}] (* Vincenzo Librandi, Apr 05 2015 *)
CoefficientList[Series[(1+2*x-x^2)/((1-x)^4*(1+x)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 05 2015 *)
PROG
(PARI) Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015
(PARI) \\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(apply(i->y, [1..30]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017
(Magma) [(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015
CROSSREFS
Column 3 of A295633.
Sequence in context: A022560 A290190 A193452 * A277029 A013934 A167189
KEYWORD
nonn
AUTHOR
EXTENSIONS
Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
First formula adapted to offset by Vaclav Kotesovec, Apr 05 2015
Name clarified by Andrew Howroyd, Nov 25 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 April 24 17:29 EDT 2024. Contains 371962 sequences. (Running on oeis4.)