cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A003444 Number of dissections of a polygon.

Original entry on oeis.org

1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

Comments

See A220881 for an essentially identical sequence, but with a different offset and a more precise definition. - N. J. A. Sloane, Dec 28 2012
Also number of necklaces of 2 colors with 2n beads and n-2 black ones. - Wouter Meeussen, Aug 03 2002

References

  • P. Lisonek, Closed forms for the number of polygon dissections. Journal of Symbolic Computation 20 (1995), 595-601.
  • R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003445, A003450, A047996, A073020, A220881.

Programs

  • Mathematica
    Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-2)/# ] &)/@Intersection[Divisors[2n], Divisors[n-2]])/(2n), {n, 3, 32}]

Formula

a(n) = (1/(2n)) Sum_{d |(2n, k)} phi(d)*binomial(2n/d, k/d) with k=n-2. - Wouter Meeussen, Aug 03 2002

Extensions

More terms from Wouter Meeussen, Aug 03 2002

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

Views

Author

Andrew Howroyd, Nov 24 2017

Keywords

Examples

			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;
...

Links

Crossrefs

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

Programs

  • PARI
    \\ 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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

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

Links

Crossrefs

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

Programs

  • Maple
    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)];
  • Mathematica
    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 *)
  • PARI
    \\ 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

Formula

See Maple program.

Extensions

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
Name clarified by Andrew Howroyd, Nov 25 2017

A003450 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 2, 6, 24, 89, 371, 1478, 6044, 24302, 98000, 392528, 1570490, 6264309, 24954223, 99253318, 394409402, 1565986466, 6214173156, 24647935156, 97732340680, 387428854374, 1535588541762, 6085702368796, 24116801236744, 95569050564444, 378718095630676
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

Comments

In other words, the number of (n - 5)-dissections of an n-gon modulo the dihedral action.
Equivalently, the number of two-dimensional faces of the (n-3)-dimensional associahedron modulo the dihedral 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

References

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

Links

Crossrefs

A diagonal of A295634.
Cf. A003449, A295419.
Cf. A003444, A003445, A220881.

Programs

  • Maple
    C:=n->binomial(2*n,n)/(n+1);
T32:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 0 then t1:=t1+((3*(n-4)*(n-1))/(16*(n-3)))*C(n/2-1) fi;
else
t1 :=  (n-3)^2*(n-4)*C(n-2)/(8*n*(2*n-5));
if n mod 5 = 0 then t1:=t1+(2/5)*C(n/5-1) fi;
if n mod 2 = 1 then t1:=t1+((n^2-2*n-11)/(8*(n-4)))*C((n-3)/2) fi;
fi;
t1; end;
[seq(T32(n),n=5..40)];
  • Mathematica
    c = CatalanNumber;
T32[n_] := Module[{t1}, If[EvenQ[n], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n*(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5)*c[n/5-1]]; If[EvenQ[n], t1 = t1 + ((3*(n-4)*(n-1))/(16*(n-3)))*c[n/2-1]], t1 = (n-3)^2*(n-4)*c[n-2]/(8*n *(2*n - 5)); If[Mod[n, 5] == 0, t1 = t1 + (2/5) * c[n/5-1]]; If[OddQ[n], t1 = t1 + ((n^2 - 2*n - 11)/(8*(n-4)))*c[(n-3)/2]]]; t1];
Table[T32[n], {n, 5, 40}] (* Jean-François Alcover, Dec 11 2017, translated from Maple *)
  • PARI
    \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(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

Formula

See Maple program.

Extensions

Entry revised (following Bowman and Regev) by N. J. A. Sloane, Dec 28 2012
Name clarified by Andrew Howroyd, Nov 24 2017

A003442 Number of nonequivalent dissections of an n-gon into (n-3) polygons by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

1, 2, 11, 48, 208, 858, 3507, 14144, 56698, 226100, 898942, 3565920, 14124496, 55887930, 220985795, 873396480, 3450940830, 13633173180, 53855628554, 212750148000, 840496068160, 3320817060132, 13122294166126, 51860761615488
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of dissections of regular n-gon into n-3 polygons without reflection and rooted at a cell. - Sean A. Irvine, May 05 2015
The conditions imposed mean that the dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 23 2017

Examples

			Case n=5: A pentagon can be dissected into 1 quadrilateral and 1 triangle. Either one of these can be chosen as the root cell so a(n)=2. - _Andrew Howroyd_, Nov 23 2017

References

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

Links

Crossrefs

Cf. A003443, A003454, A220881, A295622.

Programs

  • PARI
    DissectionsModCyclicRooted(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3,#v,x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m,r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m,vars)));
my(p=O(x*x^n) + sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d,i/d))/i)));
vector(n,i,polcoeff(p,i))}
{ my(v=DissectionsModCyclicRooted(apply(i->if(i>=3&&i<=4,y^(i-3) + O(y^2)), [1..25]))); apply(p->polcoeff(p,1), v[4..#v]) } \\ Andrew Howroyd, Nov 22 2017

Extensions

More terms from Sean A. Irvine, May 05 2015
Name clarified by Andrew Howroyd, Nov 22 2017

A220882 Number of (n - 6)-dissections of an n-gon (equivalently, the number of three-dimensional faces of the (n-3)-dimensional associahedron) modulo the cyclic action.

Original entry on oeis.org

1, 2, 16, 93, 505, 2548, 12400, 58140, 266550, 1198564, 5312032, 23263695, 100910001, 434217000, 1855972096, 7887862224, 33359979546, 140492933100, 589495272736, 2465455090098, 10281760786682, 42768958597992, 177499631598976, 735146520745000, 3039095720959424, 12542491305496152
Offset: 6

Views

Author

N. J. A. Sloane, Dec 28 2012

Keywords

Links

Crossrefs

Cf. A003444, A003445, A003450, A220881.

Programs

  • Maple
    C:=n->binomial(2*n,n)/(n+1);
T4:=proc(n) local t1; global C;
t1 :=  (((n-3)*(n-4)^2*(n-5))/(24*n*(2*n-5)))*C(n-2);
if n mod 2 = 0 then t1:=t1+((n-4)^2/(4*n))*C(n/2-2) fi;
if n mod 3 = 0 then t1:=t1+((n-3)/9)*C(n/3-1) fi;
if n mod 6 = 0 then t1:=t1+C(n/6-1)/3 fi;
t1; end;
[seq(T4(n),n=6..40)];
  • Mathematica
    c = CatalanNumber;
T4[n_] := Module[{t1},
t1 = (((n - 3)*(n - 4)^2*(n - 5))/(24*n*(2*n - 5)))*c[n - 2];
If[Mod[n, 2] == 0, t1 = t1 + ((n - 4)^2/(4*n))*c[n/2 - 2]];
If[Mod[n, 3] == 0, t1 = t1 + ((n - 3)/9)*c[n/3 - 1]];
If[Mod[n, 6] == 0, t1 = t1 + c[n/6 - 1]/3]; t1];
Table[T4[n], {n, 6, 40}] (* Jean-François Alcover, Dec 02 2017, from Maple *)

Formula

See Maple code.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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