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.

Previous Showing 11-20 of 20 results.

A002712 Number of unrooted triangulations of a disk that have reflection symmetry with n interior nodes and 3 nodes on the boundary.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 68, 215, 680, 2226, 7327, 24607, 83060, 284046, 975950, 3383343, 11778308, 41269252, 145131502, 512881550, 1818259952, 6470758289, 23091680690, 82659905947, 296605398856, 1067012168350, 3846553544904, 13896522968160, 50296815014780, 182378110257354, 662384549806938
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [n,0]-triangulations.

References

  • 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

Column k=0 of A169809.

Programs

  • Maple
    Dc := proc(n,m) 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ; end:
A000260 := proc(n) Dc(n,0) ; end:
Dx2 := proc(nmax) add( A000260(n)*x^(2*n),n=0..nmax) ; end:
o := 20: Order := 2*o-1 : j := solve( J0=1+x*J0+x^2*J0*(1+x*J0/2)*series(J0^2-Dx2(o),x=0,2*o-1),J0) ;
for n from 0 to 2*o-2 do printf("%d,",coeftayl(j,x=0,n)) ; od: # R. J. Mathar, Oct 29 2008
  • Mathematica
    seq[m_] := Module[{q}, q = Sum[x^(2n) Binomial[4n+2, n+1]/ ((2n+1)(3n+2)), {n, 0, Quotient[m, 2]}]; p = 1+O[x]; Do[p = 1 + x*p + x^2*p*(1+x*p/2)(p^2-q), {n, 1, m}]; CoefficientList[p, x]];
seq[30] (* Jean-François Alcover, Apr 25 2023, after Andrew Howroyd *)
  • PARI
    seq(n)={my(q=sum(n=0, n\2, x^(2*n)*binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))), p=1+O(x)); for(n=1, n, p = 1 + x*p + x^2*p*(1 + x*p/2)*(p^2 - q)); Vec(p)} \\ Andrew Howroyd, Feb 24 2021

Extensions

More terms from R. J. Mathar, Oct 29 2008
Name clarified and terms a(27) and beyond from Andrew Howroyd, Feb 24 2021

A005502 Number of unrooted triangulations of a hexagon with n internal nodes.

Original entry on oeis.org

3, 11, 53, 295, 1867, 12560, 89038, 652198, 4903955, 37627699, 293607612, 2323604832, 18614121391, 150704813812, 1231596828200, 10148762396401, 84252059397251, 704122279126074, 5920239345451780, 50051285956517452, 425273487358680290, 3630084126997807369
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [n,3]-triangulations.
Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P6 -c2m2 [n]". - Manfred Scheucher, Mar 08 2018

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=3 of the array in A169808.
Cf. A005507, A005495.

Formula

a(n) = (A005507(n) + A005495(n))/2 (based on Max Alekseyev's formula, cf. A005501 and A005500).

Extensions

a(5)-a(10) from Manfred Scheucher, Mar 08 2018
Name clarified and terms a(11) and beyond from Andrew Howroyd, Feb 22 2021

A005505 Number of unrooted triangulations with reflection symmetry of a quadrilateral with n internal nodes.

Original entry on oeis.org

1, 2, 4, 10, 29, 86, 266, 844, 2742, 9032, 30202, 101988, 347914, 1195500, 4138310, 14405848, 50428392, 177321636, 626250990, 2219876580, 7896651847, 28176271634, 100830069380, 361757157484, 1301092926454, 4689840961196, 16940093338162, 61305930699382
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [n,1]-triangulations.

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=1 of the array in A169809.
Cf. A002710, A005500.

Formula

a(n) = 2 * A005500(n) - A002710(n) (based on Max Alekseyev's formula, cf. A005500).

Extensions

a(7)-a(12) from Altug Alkan and Manfred Scheucher, Mar 08 2018
Name clarified and terms a(13) and beyond from Andrew Howroyd, Feb 21 2021

A005506 Number of unrooted triangulations with reflection symmetry of a pentagon with n internal nodes.

Original entry on oeis.org

1, 3, 7, 19, 57, 176, 557, 1806, 5954, 19897, 67235, 229366, 788688, 2730810, 9512107, 33309444, 117190184, 414039578, 1468349782, 5225201321, 18651958885, 66769742002, 239643164237, 862168692562, 3108716586702, 11232127258416, 40660388117380, 147453014455094
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [n,2]-triangulations.

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=2 of the array in A169809.
Cf. A002711, A005501.

Formula

a(n) = 2 * A005501(n) - A002711(n) (based on Max Alekseyev's formula, cf. A005501).

Extensions

a(6)-a(11) from Altug Alkan and Manfred Scheucher, Mar 08 2018
Name clarified and terms a(12) and beyond from Andrew Howroyd, Feb 21 2021

A005507 Number of unrooted triangulations with reflection symmetry of a hexagon with n internal nodes.

Original entry on oeis.org

2, 6, 18, 52, 166, 524, 1722, 5664, 19072, 64408, 220676, 758864, 2634734, 9180872, 32208376, 113371636, 401067522, 1423073892, 5068961452, 18103192360, 64853607912, 232872927444, 838311889890, 3023961593292, 10931277735230, 39586258360246, 143617299291242
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [n,3]-triangulations.

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=3 of the array in A169809.
Cf. A005495, A005502.

Formula

a(n) = 2 * A005502(n) - A005495(n) (based on Max Alekseyev's formula, cf. A005500 and A005501).

Extensions

a(5)-a(10) from Altug Alkan and Manfred Scheucher, Mar 08 2018
Name clarified and terms a(11) and beyond from Andrew Howroyd, Feb 21 2021

A005504 Number of unrooted triangulations of a disk with 2 internal nodes and n+3 nodes on the boundary.

Original entry on oeis.org

1, 5, 14, 53, 178, 685, 2548, 9876, 37950, 147520, 572594, 2230735, 8693932, 33939465, 132598484, 518607032, 2029990774, 7952788446, 31179668572, 122331725930, 480283816348, 1886829349570, 7416950176904, 29171683995320, 114795961678380, 451968102200966, 1780298693036010
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [2,n]-triangulations.
Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P[n] -c2m2 [n+2]". - Manfred Scheucher, Mar 08 2018

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row n=2 of the array in A169808.

Extensions

a(6)-a(12) from Manfred Scheucher, Mar 08 2018
Name clarified and terms a(13) and beyond from Andrew Howroyd, Feb 22 2021

A005508 Number of unrooted triangulations with reflection symmetry of a disk with one internal node and n+3 nodes on the boundary.

Original entry on oeis.org

1, 2, 3, 6, 8, 18, 23, 56, 70, 180, 222, 594, 726, 2002, 2431, 6864, 8294, 23868, 28730, 83980, 100776, 298452, 357238, 1069776, 1277788, 3863080, 4605980, 14040810, 16715250, 51325650, 61020495, 188574240, 223931910, 695987820, 825632610, 2579248980
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [1,n]-triangulations.

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row n=1 of the array in A169809.

Extensions

Name clarified and terms a(7) and beyond from Andrew Howroyd, Feb 22 2021

A005509 Number of unrooted triangulations with reflection symmetry of a disk with 2 internal nodes and n+3 nodes on the boundary.

Original entry on oeis.org

1, 4, 7, 18, 26, 68, 91, 248, 318, 900, 1122, 3278, 4004, 12012, 14443, 44304, 52598, 164424, 193154, 613700, 714476, 2302344, 2659582, 8677072, 9954860, 32836180, 37442160, 124715430, 141430680, 475237500, 536257995, 1816267680, 2040199590, 6959878200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row n=2 of the array in A169809.

Extensions

Terms a(6) and beyond from Andrew Howroyd, Feb 22 2021

A005503 Number of unrooted triangulations of a disk with one internal node and n+3 nodes on the boundary.

Original entry on oeis.org

1, 2, 4, 11, 28, 91, 291, 1004, 3471, 12350, 44114, 159519, 579835, 2121845, 7800702, 28813730, 106844383, 397647256, 1484755972, 5560561040, 20881939915, 78617991116, 296678132514, 1121988213996, 4251702739831, 16141719280994, 61389611762126, 233856524866209
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are also called [1,n]-triangulations.
Graphs can be enumerated and counted using the tool "plantri", in particular the command "./plantri -s -P[n] -c2m2 [n+1]". - Manfred Scheucher, Mar 08 2018

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Row n=1 of the array in A169808.

Extensions

a(6) corrected and a(7)-a(14) from Manfred Scheucher, Mar 08 2018
Name clarified and terms a(15) and beyond from Andrew Howroyd, Feb 22 2021

A169807 Erroneous version of A005028.

Original entry on oeis.org

1, 2, 4, 12, 33, 102, 312, 1006
Offset: 3

Views

Author

Keywords

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.

Links

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Previous Showing 11-20 of 20 results.

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

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