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
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, Aug 02 2017
STATUS
approved