A291055 Number of maximal irredundant sets in the n-path graph.

1, 2, 2, 4, 6, 8, 13, 17, 27, 40, 57, 86, 122, 184, 269, 395, 582, 849, 1255, 1843, 2708, 3982, 5841, 8597, 12631, 18566, 27286, 40082, 58929, 86598, 127279, 187052, 274872, 404001, 593732, 872606, 1282416, 1884660, 2769856, 4070718, 5982611, 8792345

Offset

1,2

Comments

From Andrew Howroyd, Aug 23 2017: (Start)

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 second if n is a multiple of 3, otherwise starting with the first.

The maximum size of an irredundant set, the upper irredundance number, is given by ceiling(n/2). A suitable construction for such a set is every second vertex starting with the first.

(End)

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Path Graph

Eric Weisstein's World of Mathematics, Maximal Irredundant Set

Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 1, 0, -1, -2, -1, 2, 1, 0, 0, -1).

Formula

From Andrew Howroyd, Aug 23 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*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - x^6 - 2*x^7 + 3*x^9 - x^12 - x^13)/(1 - x^2 - x^3 - x^4 - x^5 + x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + x^14).

(End)

Example

Case n=5: maximal irredundant sets represented as binary words are {00110, 01001, 01010, 01100, 10010, 10101}, so a(5)=6. - Andrew Howroyd, Aug 23 2017

Mathematica

Rest @ CoefficientList[Series[x (1 + 2 x + x^2 + x^3 + x^4 - x^5 - x^6 - 2 x^7 + 3 x^9 - x^12 - x^13)/(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, 42}], x] (* Michael De Vlieger, Aug 24 2017 *)
LinearRecurrence[{0, 1, 1, 1, 1, 0, -1, -2, -1, 2, 1, 0, 0, -1}, {1, 2, 2, 4, 6, 8, 13, 17, 27, 40, 57, 86, 122, 184}, 20] (* Eric W. Weisstein, Aug 28 2017 *)
RootSum[1 - #^3 - 2 #^4 + #^5 + 2 #^6 + #^7 - #^9 - #^10 - #^11 - #^12 + #^14 &, -4480566127993567 #^n + 2115784835595702 #^(n+1) - 8803686900182082 #^(n+2) + 12438105918248674 #^(n+3) + 9975829435558087 #^(n+4) + 32647411155695559 #^(n+5) + 921201586573742 #^(n+6) - 12400355965941932 #^(n+7) - 18709447182799197 #^(n+8) - 16194871035876814 #^(n+9) - 8478829128434826 #^(n+10) - 3824486277258301 #^(n+11) + 902031297001609 #^(n+12) + 11119370357865554 #^(n+13) &]/333325507942333403 (* Eric W. Weisstein, Aug 28 2017 *)

Prog

(PARI) Vec((1 + 2*x + x^2 + x^3 + x^4 - x^5 - x^6 - 2*x^7 + 3*x^9 - x^12 - x^13)/(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 23 2017

Crossrefs

Row 1 of A291439.

Row sums of A291375.

Cf. A286887, A286954, A291444.

Sequence in context: A274152 A274155 A145465 * A266949 A255710 A329137

Adjacent sequences: A291052 A291053 A291054 * A291056 A291057 A291058

Keyword

nonn

Author

Eric W. Weisstein, Aug 17 2017

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 23 2017

Status

approved