A286887 Number of irredundant sets in the path graph P_n.

2, 3, 5, 9, 15, 26, 44, 76, 130, 223, 382, 655, 1123, 1925, 3300, 5657, 9698, 16625, 28500, 48857, 83755, 143580, 246137, 421949, 723341, 1240013, 2125736, 3644118, 6247058, 10709240, 18358693, 31472038, 53952053, 92489213, 158552901, 271804912, 465951173

Offset

1,1

Comments

Equivalently, the number of binary words of length n that don't start or end with 11 (the outside 1 is redundant) and don't contain 111, 1101 or 1011 (the middle 1 is redundant).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Irredundant Set

Eric Weisstein's World of Mathematics, Path Graph

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

Formula

a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) for n > 6.

G.f.: x*(1 + x)*(2 - x + x^2 - x^4)/(1 - x - x^2 - x^4 + x^6).

Example

Case n=5: irredundant words are {00000, 00001, 00010, 00100, 01000, 10000, 00110, 01100, 10001, 00101, 01010, 10100, 01001, 10010, 10101}, so a(5)=15.

Mathematica

RootSum[1 - #^2 - #^4 - #^5 + #^6 &, 3191 #^n + 4752 #^(1 + n) - 4234 #^(2 + n) + 11985 #^(3 + n) - 2369 #^(4 + n) + 3536 #^(5 + n) &]/89653 (* Eric W. Weisstein, Aug 04 2017 *)
LinearRecurrence[{1, 1, 0, 1, 0, -1}, {2, 3, 5, 9, 15, 26}, 20] (* Eric W. Weisstein, Aug 04 2017 *)
CoefficientList[Series[(2 + x + x^3 - x^4 - x^5)/(1 - x - x^2 - x^4 + x^6), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 04 2017 *)

Prog

(PARI) Vec((1 + x)*(2 - x + x^2 - x^4)/(1 - x - x^2 - x^4 + x^6) + O(x^40))

Crossrefs

Row 1 of A286868 and A286870.

Sequence in context: A302018 A096816 A220127 * A018157 A228644 A003065

Adjacent sequences: A286884 A286885 A286886 * A286888 A286889 A286890

Keyword

nonn,easy

Author

Andrew Howroyd, Aug 02 2017

Status

approved