OFFSET
0,2
COMMENTS
An "overpal" is a word of the form a x a x^R a, where a is a single letter, x is a (possibly empty) word, and x^R denotes the reverse of the word x. To be "overpal-free" is to contain no factor (contiguous block) that is an overpal.
A binary word avoids overpals if and only if it avoids aaa, ababa, and abbabba as factors (Narad Rampersad). This gives the proof of Barker's formulas below. - Jeffrey Shallit, Oct 09 2016 and Colin Barker, Oct 10 2016
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Aayush Rajasekaran, Narad Rampersad, Jeffrey Shallit, Overpals, Underlaps, and Underpals, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).
FORMULA
From Colin Barker, Oct 08 2016: (Start)
a(n) = a(n-2)+a(n-4) for n>9.
G.f.: (1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+6*x^6+8*x^7+6*x^8+2*x^9) / (1-x^2-x^4).
(End)
EXAMPLE
For n = 4, the 14 words are 00100, 00101, 00110, 01001, and their complements and reversals.
PROG
(PARI) Vec((1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+6*x^6+8*x^7+6*x^8+2*x^9)/(1-x^2-x^4) + O(x^50)) \\ Colin Barker, Oct 10 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jeffrey Shallit, Oct 08 2016
STATUS
approved