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 1-10 of 11 results. Next

A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 3, 9, 20, 75, 262, 1117, 4783, 21971, 102249, 489077, 2370142, 11654465, 57916324, 290693391, 1471341341, 7504177738, 38532692207, 199076194985, 1034236705992, 5400337050086, 28329240333758, 149244907249629
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Original name: number of symmetric dissections of a polygon.
Also number of 2-connected outerplanar graphs on n unlabeled nodes. - Steven Finch, Dec 09 2004

References

  • Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014
  • Guanzhang Hu, Group theory method for enumeration of outerplanar graphs, Acta Math. Appl. Sinica 14 (1998) 381-387.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003454, A003455, A003456, A005036, A295260.

Programs

  • Mathematica
    f[x_, n_]:=x+Sum[(1/r)*Binomial[s-2, r-1]*Binomial[r+s-1, s]*x^s, {r, 1, n}, {s, 2, n}]; F[x_, n_]:=Series[((3x^2-2*x*f[x, n]+f[x, n]^2)- (2+2*x+7*x^2-4*x*f[x, n]+2*f[x, n]^2)*f[x^2, n]+ 2*f[x^2, n]^2)/(4*(2*f[x^2, n]-1))+Sum[If[Mod[k, d]==0, EulerPhi[d]*f[x^d, n]^(k/d)/k, 0], {k, 3, n}, {d, 1, k}]/2, {x, 0, n}]; F[x, 22] (Finch)
  • PARI
    \\ See A295419 for DissectionsModDihedral().
my(v=DissectionsModDihedral(apply(i->1, [1..30])));v[3..#v] \\ Andrew Howroyd, Nov 22 2017

Extensions

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 21 2005
Name clarified by Andrew Howroyd, Nov 22 2017

A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 6, 7, 4, 1, 3, 11, 24, 24, 12, 1, 3, 17, 51, 89, 74, 27, 1, 4, 26, 109, 265, 371, 259, 82, 1, 4, 36, 194, 660, 1291, 1478, 891, 228, 1, 5, 50, 345, 1477, 3891, 6249, 6044, 3176, 733, 1, 5, 65, 550, 3000, 10061, 21524, 29133, 24302, 11326, 2282
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,  3,   3;
1, 2,  6,   7,   4;
1, 3, 11,  24,  24,   12;
1, 3, 17,  51,  89,   74,   27;
1, 4, 26, 109, 265,  371,  259,  82;
1, 4, 36, 194, 660, 1291, 1478, 891, 228;
...

Links

Crossrefs

Row sums are A001004.
Column k=3 is A003453.
Diagonals include A000207, A003449, A003450.
Cf. A295260, A295419, A295633.

Programs

  • PARI
    \\ See A295419 for DissectionsModDihedral()
T=DissectionsModDihedral(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

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

A003453 Number of nonequivalent dissections of an n-gon into 3 polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 3, 6, 11, 17, 26, 36, 50, 65, 85, 106, 133, 161, 196, 232, 276, 321, 375, 430, 495, 561, 638, 716, 806, 897, 1001, 1106, 1225, 1345, 1480, 1616, 1768, 1921, 2091, 2262, 2451, 2641, 2850, 3060, 3290, 3521, 3773, 4026
Offset: 5

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

In other words, the number of 2-dissections of an n-gon modulo the dihedral action.
John W. Layman observes that this appears to be the alternating sum transform (PSumSIGN) of A005744.
Row 2 of the convolution array A213847. - Clark Kimberling, Jul 05 2012
Number of nonisomorphic outer planar graphs of order n >= 3 and size n+2. - Christian Barrientos and Sarah Minion, Feb 27 2018

References

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

Links

Crossrefs

Column 3 of A295634.
Cf. A005744, A213847, A295419.

Programs

  • Maple
    T52:= proc(n)
if n mod 2 = 0 then (n-4)*(n-2)*(n+3)/24;
else (n-3)*(n^2-13)/24; fi end;
[seq(T52(n),n=5..80)]; # N. J. A. Sloane, Dec 28 2012
  • Mathematica
    nd[n_]:=If[EvenQ[n],(n-4)(n-2) (n+3)/24,(n-3) (n^2-13)/24]; Array[nd,50,5] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{1,3,6,11,17,26},50] (* Harvey P. Dale, Jan 28 2013 *)
  • PARI
    \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(apply(i->y + O(y^4), [1..40]))); apply(p->polcoeff(p, 3), v[5..#v]) } \\ Andrew Howroyd, Nov 24 2017

Formula

G.f.: (1+x-x^2) / ((1-x)^4*(1+x)^2).
See also the Maple code.
a(5)=1, a(6)=3, a(7)=6, a(8)=11, a(9)=17, a(10)=26, a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a (n-6). - Harvey P. Dale, Jan 28 2013
a(n) = (2*n^3-6*n^2-23*n+63+3*(n-5)*(-1)^n)/48, for n>=5. - Luce ETIENNE, Apr 07 2015
a(n) = (1/2) * Sum_{i=1..n-4} floor((i+1)*(n-i-2)/2). - Wesley Ivan Hurt, May 07 2016

Extensions

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

A295495 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations.

Original entry on oeis.org

1, 1, 2, 5, 11, 36, 114, 410, 1458, 5488, 20786, 80770, 317378, 1265139, 5094139, 20718347, 84961256, 351086326, 1460591637, 6113826319, 25733864299, 108867782794, 462707558813, 1974991841442, 8463121111860, 36397780088126, 157066702354947, 679917566925030
Offset: 3

Views

Author

Andrew Howroyd, Nov 22 2017

Keywords

Links

Crossrefs

Cf. A003455, A295224, A295419.

Programs

  • Mathematica
    DissectionsModCyclic[v_] :=
Module[{n = Length[v], q, vars, u, p}, q = Table[0, {n}]; q[[1]] = InverseSeries[x - Sum[x^i v[[i]], {i, 3, Length[v]}]/x + O[x]^(n+1)]; For[i = 2, i <= n, i++, q[[i]] = q[[i-1]] q[[1]]]; vars = Variables[q[[1]]]; u[m_, r_] := Normal[(q[[r]] + O[x]^(Quotient[n, m] + 1))] /. Thread[vars -> vars^m]; p = O[x]^n + x u[1, 1] - x^2 + (u[2, 1] - u[1, 2])/2 + Sum[v[[i]] Sum[EulerPhi[d] u[d, i/d]/i, {d, Divisors[i]}], {i, 3, Length[v]}]; Drop[CoefficientList[p, x], 3]];
DissectionsModCyclic[Boole[PrimeQ[#]]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 26 2019, after Andrew Howroyd *)
  • PARI
    \\ number of dissections into parts defined by set.
DissectionsModCyclic(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) + x*u(1,1) - x^2 + (u(2,1)-u(1,2))/2 + 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))}
DissectionsModCyclic(apply(i->isprime(i), [1..30]))

A003449 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 3, 7, 24, 74, 259, 891, 3176, 11326, 40942, 148646, 543515, 1996212, 7367075, 27294355, 101501266, 378701686, 1417263770, 5318762098, 20011847548, 75473144396, 285267393358, 1080432637662, 4099856060808, 15585106611244, 59343290815356
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

Comments

In other words, the number of almost-triangulations of an n-gon modulo the dihedral action.
Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the dihedral action.
The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 24 2017
See Theorem 30 of Bowman and Regev (although there appears to be a typo in the formula - see Maple code below). - N. J. A. Sloane, Dec 28 2012

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. A003450, A295419.

Programs

  • Maple
    C:=n->binomial(2*n,n)/(n+1);
T30:=proc(n) local t1; global C;
if n mod 2 = 0 then
t1:=(1/4-(3/(4*n)))*C(n-2) + (3/8)*C(n/2-1) + (1-3/n)*C(n/2-2);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/4 fi;
else
t1:=(1/4-(3/(4*n)))*C(n-2) + (1/2)*C((n-3)/2);
fi;
t1; end;
[seq(T30(n),n=4..40)]; # N. J. A. Sloane, Dec 28 2012
  • Mathematica
    c = CatalanNumber;
T30[n_] := Module[{t1}, If[EvenQ[n], t1 = (1/4 - (3/(4*n)))*c[n - 2] + (3/8)*c[n/2 - 1] + (1 - 3/n)*c[n/2 - 2]; If[Mod[n, 4] == 0, t1 = t1 + c[n/4 - 1]/4], t1 = (1/4 - (3/(4*n)))*c[n-2] + (1/2)*c[(n-3)/2]]; t1];
Table[T30[n], {n, 4, 40}] (* Jean-François Alcover, Dec 14 2017, after N. J. A. Sloane *)
  • PARI
    \\ See A295419 for DissectionsModDihedral()
{ my(v=DissectionsModDihedral(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 24 2017

Extensions

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

A290646 Number of dissections of an n-gon into 3- and 4-gons counted up to rotations and reflections.

Original entry on oeis.org

1, 2, 2, 7, 14, 53, 171, 691, 2738, 11720, 50486, 224012, 1005468, 4581815, 21093190, 98093226, 459986674, 2173599817, 10340539744, 49496519950, 238240366274, 1152543685463, 5601603835982, 27341242042238, 133977037982121, 658902522544060, 3251446102879398
Offset: 3

Views

Author

Evgeniy Krasko, Sep 03 2017

Keywords

Examples

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one triangle and one quadrangle.

Links

Crossrefs

Cf. A001004 (counted distinctly).
Cf. A001002, A290571, A295260, A295419.

Programs

  • Mathematica
    (* See A295419 for DissectionsModDihedral. *)
DissectionsModDihedral[Boole[# == 3 || # == 4]& /@ Range[1, 30]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)
  • PARI
    \\ See A295419 for DissectionsModDihedral.
DissectionsModDihedral(apply(v->v==3||v==4, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Extensions

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A290722 Number of dissections of a 2n-gon into polygons with even number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 2, 4, 13, 48, 238, 1325, 8297, 54519, 373363, 2621872, 18797682, 136969519, 1011903735, 7564219361, 57129086391, 435394899361, 3345082819597, 25885718422329, 201619294539406, 1579629974876090, 12442262963919863, 98483477967355109, 783017782731507416
Offset: 2

Views

Author

Evgeniy Krasko, Sep 03 2017

Keywords

Examples

			For a(4) = 4 the dissections of an octagon are: two dissections into 3 quadrangles; a dissection into one hexagon and one quadrangle; a dissection into one octagon.

Links

Crossrefs

Cf. A003168 (counted distinctly).
Cf. A001004, A290816, A295419.

Programs

  • PARI
    \\ See A295419 for DissectionsModDihedral().
select(v->v>0, DissectionsModDihedral(apply(v->v%2==0, [1..50]))) \\ Andrew Howroyd, Nov 22 2017

Extensions

Terms a(8) and beyond from Andrew Howroyd, Nov 22 2017

A290571 Number of dissections of an n-gon into 3- and 5-gons counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 7, 22, 60, 208, 695, 2566, 9451, 36158, 139574, 548347, 2174801, 8719651, 35244472, 143581782, 588858667, 2430036786, 10083626092, 42055927173, 176217259551, 741517642476, 3132564196880, 13281805256068, 56503895845238, 241135999611542
Offset: 3

Views

Author

Evgeniy Krasko, Sep 03 2017

Keywords

Examples

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Links

Crossrefs

Cf. A001004, A290646, A295260.

Programs

  • PARI
    \\ See A295419 for DissectionsModDihedral().
DissectionsModDihedral(apply(v->v==3||v==5, [1..25])) \\ Andrew Howroyd, Nov 22 2017

Extensions

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017

A290816 Number of dissections of an n-gon into polygons with odd number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 8, 23, 65, 223, 757, 2824, 10559, 40994, 160734, 641420, 2584587, 10528305, 43237978, 178974779, 745814185, 3127246179, 13185588894, 55878618492, 237905685582, 1017225981255, 4366536472758, 18812074137141, 81320795918871, 352638701880227
Offset: 3

Views

Author

Evgeniy Krasko, Sep 03 2017

Keywords

Examples

			For a(5) = 2 the dissections of a pentagon are: a dissection into 3 triangles; a dissection into one pentagon.

Links

Crossrefs

Cf. A049124 (counted distinctly).
Cf. A001004, A290722, A295419.

Programs

Extensions

Terms a(16) and beyond from Andrew Howroyd, Nov 22 2017
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