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.

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

Original entry on oeis.org

426, 4968, 50640, 486930, 4547088, 41796168, 380789562, 3451622904, 31194607488, 281440825122, 2536622917920, 22848990484344, 205743704494026, 1852238413383048, 16673036119790640, 150072652217086770, 1350735146332489008, 12157047307392618408
Offset: 3

Views

Author

Peter Kagey, Oct 13 2020

Keywords

Comments

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

Examples

			For n = 3, the 3-antiprism is the octahedron (3-dimensional cross-polytope), so a(3) = A033815(3) = 426.

Links

Crossrefs

Cf. A077263, A124352, A124353, A192742, A284699, A284700, A284701, A287988, A294152, A297384.
Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338152 (demihypercube), A338153 (prism).

Programs

  • Mathematica
    A338154[n_] := Round[-2^(1-n)*((7 - Sqrt[13])^n + (7 + Sqrt[13])^n) + 9^n + 5] (* Peter Kagey, Nov 15 2020 *)

Formula

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 6*x^3*(71 - 379*x + 612*x^2 - 324*x^3) / ((1 - x)*(1 - 9*x)*(1 - 7*x + 9*x^2)).
a(n) = 17*a(n-1) - 88*a(n-2) + 153*a(n-3) - 81*a(n-4) for n>6.
(End)
a(n) = -2^(1-n)*((7-sqrt(13))^n + (7+sqrt(13))^n) + 9^n + 5. - Peter Kagey, Nov 15 2020

A284701 Number of maximal matchings in the n-antiprism graph.

Original entry on oeis.org

2, 6, 14, 46, 137, 354, 905, 2366, 6278, 16681, 44156, 116650, 308180, 814645, 2153984, 5695102, 15056494, 39804582, 105231559, 278204561, 735502187, 1944477640, 5140687360, 13590620330, 35930023287, 94989547620, 251127430313, 663914974741
Offset: 1

Views

Author

Eric W. Weisstein, Apr 01 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 16 2017

Links

Crossrefs

Cf. A284703, A284710, A192742, A284700.

Programs

  • Mathematica
    LinearRecurrence[{2, 1, 0, 3, 5, 1, -2, -1}, {2, 6, 14, 46, 137, 354,
  905, 2366}, 20] (* Eric W. Weisstein, May 17 2017 *)
CoefficientList[Series[x*(-8*x^7-14*x^6+6*x^5+25*x^4+12*x^3+2*x+2)/(x^8 +2*x^7-x^6-5*x^5 -3*x^4-x^2-2*x+1), {x, 0, 50}], x] (* G. C. Greubel, May 17 2017 *)
Table[RootSum[1 + 2 # - #^2 - 5 #^3 - 3 #^4 - #^6 - 2 #^7 + #^8 &, #^n &], {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
  • PARI
    Vec((-8*x^7-14*x^6+6*x^5+25*x^4+12*x^3+2*x+2)/(x^8+2*x^7-x^6-5*x^5-3*x^4-x^2-2*x+1)+O(x^20)) \\ Andrew Howroyd, May 16 2017

Formula

From Andrew Howroyd, May 16 2017 (Start)
a(n) = 2*a(n-1) + a(n-2) + 3*a(n-4) + 5*a(n-5) + a(n-6) - 2*a(n-7) - a(n-8) for n>8.
G.f.: x*(-8*x^7 - 14*x^6 + 6*x^5 + 25*x^4 + 12*x^3 + 2*x + 2)/(x^8 + 2*x^7 - x^6 - 5*x^5 - 3*x^4 - x^2 - 2*x + 1). (End)

Extensions

a(1)-a(2) and a(16)-a(28) from Andrew Howroyd, May 16 2017

A286910 Number of independent vertex sets and vertex covers in the n-antiprism graph.

Original entry on oeis.org

3, 1, 5, 10, 21, 46, 98, 211, 453, 973, 2090, 4489, 9642, 20710, 44483, 95545, 205221, 440794, 946781, 2033590, 4367946, 9381907, 20151389, 43283149, 92967834, 199685521, 428904338, 921243214, 1978737411, 4250128177, 9128846213, 19607839978, 42115660581
Offset: 0

Views

Author

Andrew Howroyd, May 15 2017

Keywords

Comments

Sequence extrapolated to n=0 using recurrence.

Links

Crossrefs

Cf. A051927, A182143, A192742, A284700.

Programs

  • Magma
    I:=[3,1,5]; [n le 3 select I[n] else Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..33]]; // Vincenzo Librandi, May 16 2017
  • Mathematica
    CoefficientList[Series[(- 2 x^2 - 2 x + 3) / (- x^3 - 2 x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, May 16 2017 *)
LinearRecurrence[{1, 2, 1}, {3, 1, 5}, 40] (* Vincenzo Librandi, May 16 2017 *)
Table[RootSum[-1 - 2 # - #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, Aug 16 2017 *)
RootSum[-1 - 2 # - #^2 + #^3 &, #^Range[20] &] (* Eric W. Weisstein, Aug 16 2017 *)
  • PARI
    Vec((-2*x^2 - 2*x + 3)/(-x^3 - 2*x^2 - x + 1)+O(x^30))

Formula

a(n) = a(n-1) + 2*a(n-2) + a(n-3) for n>=3.
G.f.: (2*x^2 + 2*x - 3)/(x^3 + 2*x^2 + x - 1).
a(n) = n*Sum_{k=1..n} C(2*k,n-k)/k, a(0)=3. - Vladimir Kruchinin, Jun 13 2020

A290698 Number of minimal edge covers in the n-antiprism graph.

Original entry on oeis.org

2, 14, 74, 286, 1157, 4778, 19623, 80478, 330293, 1355629, 5563527, 22832914, 93707772, 384582275, 1578347684, 6477630782, 26584574434, 109104640685, 447771795953, 1837681518261, 7541951930181, 30952609765223, 127031312347552, 521343900861138
Offset: 1

Views

Author

Eric W. Weisstein, Aug 09 2017

Keywords

Comments

The n-antiprism graph is well defined for n >= 3. Sequence extended to n = 1 using recurrence. - Andrew Howroyd, Aug 10 2017

Links

Crossrefs

Cf. A284700.

Programs

  • Mathematica
    Table[2 Cos[2 n Pi/3] + RootSum[-1 + 2 # + 3 #^2 - 2 #^3 - 6 #^4 - 3 #^5 - 3 #^6 + #^7 &, #^n &], {n, 20}]
LinearRecurrence[{2, 5, 12, 11, 5, -3, -4, -1, 1}, {2, 14, 74, 286, 1157, 4778, 19623, 80478, 330293}, 20]
CoefficientList[Series[(-2 - 10 x - 36 x^2 - 44 x^3 - 25 x^4 + 18 x^5 + 28 x^6 + 8 x^7 - 9 x^8)/(-1 + 2 x + 5 x^2 + 12 x^3 + 11 x^4 + 5 x^5 - 3 x^6 - 4 x^7 - x^8 + x^9), {x, 0, 20}], x]
  • PARI
    Vec((2 + 10*x + 36*x^2 + 44*x^3 + 25*x^4 - 18*x^5 - 28*x^6 - 8*x^7 + 9*x^8)/((1 - 3*x - 3*x^2 - 6*x^3 - 2*x^4 + 3*x^5 + 2*x^6 - x^7)*(1 + x + x^2)) + O(x^30)) \\ Andrew Howroyd, Aug 10 2017

Formula

From Andrew Howroyd, Aug 10 2017: (Start)
a(n) = 2*a(n-1) + 5*a(n-2) + 12*a(n-3) + 11*a(n-4) + 5*a(n-5) - 3*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
G.f.: x*(2 + 10*x + 36*x^2 + 44*x^3 + 25*x^4 - 18*x^5 - 28*x^6 - 8*x^7 + 9*x^8)/((1 - 3*x - 3*x^2 - 6*x^3 - 2*x^4 + 3*x^5 + 2*x^6 - x^7)*(1 + x + x^2)).
(End)

Extensions

a(1)-a(2) and terms a(7) and beyond from Andrew Howroyd, Aug 10 2017
Showing 1-4 of 4 results.

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