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 1-4 of 4 results.

A220881 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 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, Dec 28 2012

Keywords

Comments

This is almost identical to A003444, but has a different offset and a more precise definition.
In other words, the number of almost-triangulations of an n-gon modulo the cyclic action.
Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the cyclic action.
The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 25 2017
Also number of necklaces of 2 colors with 2n-4 beads and n black ones. - Wouter Meeussen, Aug 03 2002

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, A003445.
Cf. A003442, A003447, A003449.

Programs

  • Maple
    C:=n->binomial(2*n,n)/(n+1);
T2:= proc(n) local t1; global C;
t1 :=  (n-3)*C(n-2)/(2*n);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/2 fi;
if n mod 2 = 0 then t1:=t1+C(n/2-1)/4 fi;
t1; end;
[seq(T2(n),n=4..40)];
  • Mathematica
    c[n_] := Binomial[2*n, n]/(n+1);
T2[n_] := Module[{t1}, t1 = (n-3)*c[n-2]/(2*n); If[Mod[n, 4] == 0, t1 = t1 + c[n/4-1]/2]; If[Mod[n, 2] == 0, t1 = t1 + c[n/2-1]/4]; t1];
Table[T2[n], {n, 4, 40}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *)
a[n_] := Sum[EulerPhi[d]*Binomial[(2n-4)/d, n/d], {d, Divisors[GCD[2n-4, n] ]}]/(2n-4);
Array[a, 30, 4] (* Jean-François Alcover, Dec 02 2017, after Andrew Howroyd *)
  • PARI
    a(n) = if(n>=4, sumdiv(gcd(2*n-4, n), d, eulerphi(d)*binomial((2*n-4)/d, n/d))/(2*n-4)) \\ Andrew Howroyd, Nov 25 2017

Formula

a(n) = (1/(2n-4)) Sum_{d |(2n-4, n)} phi(d)*binomial((2n-4)/d, n/d) for n >= 4. - Wouter Meeussen, Aug 03 2002

Extensions

Name clarified by Andrew Howroyd, Nov 25 2017

A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 2, 5, 17, 62, 275, 1272, 6225, 31075, 158376, 816229, 4251412, 22319056, 117998524, 627573216, 3355499036, 18025442261, 97239773408, 526560862829, 2861189112867, 15595669996482, 85252072993968, 467247847612316, 2567091151780343
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

Total number of dissections of an n-gon into polygons with reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

References

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

Links

Crossrefs

Cf. A001004, A003454, A003455, A005035, A295259.

Programs

  • PARI
    \\ See A003447 for DissectionsModDihedralRooted()
DissectionsModDihedralRooted(apply(i->1, [1..30]))

Extensions

More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 24 2017
a(15) corrected by Andrew Howroyd, Nov 24 2017

A003448 Number of nonequivalent dissections of an n-gon into n-4 polygons by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 3, 15, 81, 422, 2124, 10223, 47813, 218130, 977354, 4315130, 18833538, 81424236, 349303352, 1488748719, 6310303727, 26621551418, 111854042306, 468309841090, 1954642186302, 8136002036672, 33782928166668, 139971138117190, 578803145957026
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of dissections of regular n-gon into n-4 polygons with reflection and rooted at a cell.- Sean A. Irvine, May 13 2015
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

Cf. A003447.

Programs

  • PARI
    \\ See A003447 for DissectionsModDihedralRooted()
my(v=DissectionsModDihedralRooted(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

Extensions

More terms from Sean A. Irvine, May 13 2015
Name clarified by Andrew Howroyd, Nov 24 2017

A003452 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

2, 7, 15, 28, 45, 69, 98, 136, 180, 235, 297, 372, 455, 553, 660, 784, 918, 1071, 1235, 1420, 1617, 1837, 2070, 2328, 2600, 2899, 3213, 3556, 3915, 4305, 4712, 5152, 5610, 6103, 6615, 7164, 7733, 8341, 8970, 9640, 10332, 11067, 11825, 12628
Offset: 5

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of dissections of regular n-gon into 3 polygons with reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

References

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

Links

Crossrefs

Cf. A003447.

Programs

  • PARI
    \\ See A003447 for DissectionsModDihedralRooted()
{ my(v=DissectionsModDihedralRooted(apply(i->y+O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

Extensions

More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 24 2017
Showing 1-4 of 4 results.

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

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