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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

Table of n, a(n) for n=5..52.

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.

C. R. 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.

Cf. A003453, A006584, A027656.

Sequence in context: A022560 A290190 A193452 * A277029 A013934 A167189

Adjacent sequences:  A003448 A003449 A003450 * A003452 A003453 A003454

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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

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Last modified October 16 13:51 EDT 2019. Contains 328093 sequences. (Running on oeis4.)