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 1-3 of 3 results.

A077265 Number of cycles in the n-th order prism graph.

Original entry on oeis.org

14, 28, 52, 94, 170, 312, 584, 1114, 2158, 4228, 8348, 16566, 32978, 65776, 131344, 262450, 524630, 1048956, 2097572, 4194766, 8389114, 16777768, 33555032, 67109514, 134218430, 268436212, 536871724, 1073742694, 2147484578, 4294968288, 8589935648, 17179870306
Offset: 3

Views

Author

Eric W. Weisstein, Nov 01 2002

Keywords

Comments

Also the number of cycles in the n-th order web graph. - Eric W. Weisstein, Dec 17 2013
Also the number of minimal edge cuts in the n-dipyramidal graph. - Eric W. Weisstein, Oct 30 2024
A subsequence of A290699.

Links

Crossrefs

Cf. A077263, A260699.
Cf. A000079, A002378, A132109.

Programs

Formula

a(n) = 2^n+n*(n-1). - Eric W. Weisstein, Dec 16 2013
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - Colin Barker, May 06 2014
G.f.: -2*x^3*(6*x^3-19*x^2+21*x-7) / ((x-1)^3*(2*x-1)). - Colin Barker, May 06 2014
a(n) = A000079(n) + A002378(n-1). - Wesley Ivan Hurt, May 07 2014
a(n) = 2*A132109(n-1). - R. J. Mathar, May 23 2016

Extensions

More terms from Eric W. Weisstein, Dec 16 2013

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

A287988 Number of (undirected) paths in the n-antiprism graph.

Original entry on oeis.org

2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640, 31292844, 115494312, 423283562, 1542120664, 5589611460, 20170172896, 72499928322, 259692909048, 927342338956, 3302291258200, 11730149911914, 41572470711288, 147031327493572, 519029653663056
Offset: 1

Views

Author

Eric W. Weisstein, Jun 03 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Jun 05 2017

Links

Crossrefs

Cf. A077263, A124352, A124353, A287992.

Programs

  • Mathematica
    Table[n RootSum[-1 - # - 3 #^2 + #^3 &, 23 #^n + 32 #^(n + 1) + 5 #^(n + 2) &]/44 - 7 n - 3 n^2 - 2 n^3, {n, 20}]
LinearRecurrence[{10, -37, 64, -58, 36, -26, 16, -5, 2, -1}, {2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640}, 20]
CoefficientList[Series[(2 (2 x^6 + 4 x^5 + x^4 + 24 x^3 - 4 x^2 + 20 x + 1) (1 - 2 x - x^2))/((1 - x)^4 (1 - 3 x - x^2 - x^3)^2), {x, 0, 20}], x]
  • PARI
    Vec(2*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1)*(1-2*x-x^2)/((1-x)^4*(1-3*x-x^2-x^3)^2) + O(x^20)) \\ Andrew Howroyd, Jun 05 2017

Formula

From Andrew Howroyd, Jun 05 2017 (Start)
a(n) = 10*a(n-1)-37*a(n-2)+64*a(n-3) -58*a(n-4)+36*a(n-5)-26*a(n-6) +16*a(n-7)-5*a(n-8) +2*a(n-9)-a(n-10) for n>10.
G.f.: 2*x*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1) * (1-2*x-x^2) / ((1-x)^4 * (1-3*x-x^2-x^3)^2).
(End)

Extensions

a(1)-a(2) and a(14)-a(24) from Andrew Howroyd, Jun 05 2017
Showing 1-3 of 3 results.

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