A295495 -id:A295495 - cp's OEIS frontend

A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

1, 2, 3, 11, 29, 122, 479, 2113, 9369, 43392, 203595, 975563, 4736005, 23296394, 115811855, 581324861, 2942579633, 15008044522, 77064865555, 398150807179, 2068470765261, 10800665952376, 56658467018647, 298489772155137, 1578702640556193

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons without reflection. - Sean A. Irvine, May 15 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 3..200
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.

Cf. A001004, A003454, A003456, A005034, A295224, A295495.

\\ See A295495 for DissectionsModCyclic().
DissectionsModCyclic(apply(v->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

More terms from Sean A. Irvine, May 15 2015

Name clarified by Andrew Howroyd, Nov 22 2017

A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 2, 8, 12, 6, 1, 3, 16, 40, 43, 19, 1, 3, 25, 93, 165, 143, 49, 1, 4, 40, 197, 505, 712, 504, 150, 1, 4, 56, 364, 1274, 2548, 2912, 1768, 442, 1, 5, 80, 646, 2878, 7672, 12400, 11976, 6310, 1424, 1, 5, 105, 1050, 5880, 19992, 42840, 58140, 48450, 22610, 4522

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  4,   4;
1, 2,  8,  12,    6;
1, 3, 16,  40,   43,   19;
1, 3, 25,  93,  165,  143,   49;
1, 4, 40, 197,  505,  712,  504,  150;
1, 4, 56, 364, 1274, 2548, 2912, 1768, 442;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A003455.
Column k=3 is A003451.
Diagonals include A001683, A220881, A003445, A220882.
Cf. A295224, A295495, A295634.

\\ See A295495 for DissectionsModCyclic()
T=DissectionsModCyclic(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 2, 8, 40, 165, 712, 2912, 11976, 48450, 195580, 784504, 3139396, 12526605, 49902440, 198499200, 788795924, 3131945190, 12428258796, 49295766000, 195464345440, 774857314042, 3071175790232, 12171403236288, 48233597481200, 191138095393700, 757436171945952

Offset: 5

N. J. A. Sloane

nonn

In other words, the number of (n-5)-dissections of an n-gon modulo the cyclic action.
Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the cyclic action.
The dissection will always be composed of either 1 pentagon and n-5 triangles or 2 quadrilaterals and n-6 triangles. - Andrew Howroyd, Nov 24 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 5..200
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.

A diagonal of A295633.
Cf. A003444, A003450, A220881.
Cf. A003443, A003448, A003450.

C:=n->binomial(2*n,n)/(n+1);
T31:=proc(n) local t1; global C;
t1 :=  (n-3)^2*(n-4)*C(n-2)/(4*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(4/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+(n-4)*C(n/2-1)/8 fi;
t1; end;
[seq(T31(n),n=5..40)];

Table[t1 = (n - 3)^2*(n - 4)*CatalanNumber[n - 2]/(4*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (4/5)*CatalanNumber[n/5 - 1]]; If[Mod[n, 2] == 0, t1 = t1 + (n - 4)*CatalanNumber[n/2 - 1]/8]; t1, {n, 5, 20}] (* T. D. Noe, Jan 03 2013 *)

\\ See A295495 for DissectionsModCyclic()
{ my(v=DissectionsModCyclic(apply(i->if(i>=3&&i<=5, y^(i-3) + O(y^3)), [1..30]))); apply(p->polcoeff(p, 2), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

See Maple program.

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012

Name clarified by Andrew Howroyd, Nov 25 2017

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

Original entry on oeis.org

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

N. J. A. Sloane

nonn

In other words, the number of 2-dissections of an n-gon modulo the cyclic action.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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).

Column 3 of A295633.

Cf. A003453, A006584, A027656.

[(n-4)*(2*n^2-4*n-3*(1-(-1)^n))/24: n in [5..60]]; // Vincenzo Librandi, Apr 05 2015

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

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 *)

Vec((1 + 2*x - x^2 ) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Michel Marcus, Apr 04 2015

\\ 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

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

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

cp's OEIS Frontend

A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

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A295633 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation.

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A003445 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation.

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

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