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 1-1 of 1 results.

A286954 Number of maximal irredundant sets in the n-cycle graph.

Original entry on oeis.org

0, 2, 3, 6, 10, 11, 14, 14, 30, 62, 66, 99, 143, 212, 343, 478, 697, 992, 1501, 2246, 3251, 4776, 6969, 10283, 15210, 22297, 32727, 47984, 70644, 103961, 152706, 224382, 329508, 484451, 712274, 1046736, 1538164, 2260109, 3321932, 4882574, 7175738, 10545581
Offset: 1

Views

Author

Eric W. Weisstein, Aug 15 2017

Keywords

Comments

From Andrew Howroyd, Aug 16 2017: (Start)
Equivalently, the number of binary words not cyclically containing a sub-word of 111, 1101, 1011, 00000, 000010, 010000, 000100, 001000, 0100010.
For n > 1, the minimum size of a maximal irredundant set, the irredundance number, is given by ceiling(n/3). A suitable construction for such a set is every third vertex starting with the first.
The maximum size of an irredundant set, the upper irredundance number, is given by floor(n/2). For n > 1, a suitable construction for such a set is every second vertex starting with the second.
(End)

Examples

			From _Andrew Howroyd_, Aug 16 2017: (Start)
Case n=2: admissible words are 01 and 10, so a(2)=2.
Case n=3: admissible words are 001, 010, 100, so a(3)=3.
Case n=7: up to rotation, admissible words are 0010011 and 0010101, so a(7) = 7*2 = 14.
(End)

Links

Crossrefs

Row sums of A291044.
Cf. A290493, A291048.

Programs

  • Mathematica
    LinearRecurrence[{0, 1, 1, 1, 1, 0, -1, -2, -1, 2, 1, 0, 0, -1}, {0, 2, 3, 6, 10, 11, 14, 14, 30, 62, 66, 99, 143, 212}, 20]
CoefficientList[Series[(x (2 + 3 x + 4 x^2 + 5 x^3 - 7 x^5 - 16 x^6 - 9 x^7 + 20 x^8 + 11 x^9 - 14 x^12))/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2 x^8 + x^9 - 2 x^10 - x^11 + x^14), {x, 0, 20}], x]
Table[RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^n &], {n, 20}]
RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, #^Range[20] &]
  • PARI
    Vec((2 + 3*x + 4*x^2 + 5*x^3 - 7*x^5 - 16*x^6 - 9*x^7 + 20*x^8 + 11*x^9 - 14*x^12)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14)+O(x^40)) \\ Andrew Howroyd, Aug 16 2017

Formula

From Andrew Howroyd, Aug 16 2017: (Start)
a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) - a(n-7) - 2*a(n-8) - a(n-9) + 2*a(n-10) + a(n-11) - a(n-14) for n > 14.
G.f.: x^2*(2 + 3*x + 4*x^2 + 5*x^3 - 7*x^5 - 16*x^6 - 9*x^7 + 20*x^8 + 11*x^9 - 14*x^12)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14).
a(p) = p * A291048(p) for p prime.
(End)

Extensions

a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, Aug 16 2017
Showing 1-1 of 1 results.

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

Content is available under The OEIS End-User License Agreement.