A003451 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation.

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

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

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.
• Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Crossrefs

Column 3 of A295633.

Cf. A003453, A006584, A027656.

Programs

Magma

[(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 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 *)

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

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

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