OFFSET
3,1
COMMENTS
If the offsets are modified, A124720 to A124723 are the 2nd to 5th Witt transform of A000079 [Moree]. - R. J. Mathar, Nov 08 2008
a(n+2) is the number of distinct unordered pairs of binary words having a total length of n letters: a(2+2) = 5 because we have the unordered pairs: (e,00),(e,01), (e,10), (e,11), (0,1) where e represents the empty word. Each pair has a total of 2 letters and the two elements of each pair are distinct words. - Geoffrey Critzer, Feb 28 2013
LINKS
Colin Barker, Table of n, a(n) for n = 3..1000
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From R. J. Mathar, Nov 08 2008]
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).
FORMULA
G.f.: x^3*(2-3 x)/((1-2 x^2)(1- 2x)^2) = (x^2/(1-2x)^2 - x^2/(1-2*x^2))/2.
From Colin Barker, Oct 28 2016: (Start)
a(n) = 2^(n-3)*(n-1)-2^(n/2-2) for n even.
a(n) = 2^(n-3)*n-2^(n-3) for n odd.
a(n) = 4*a(n-1)-2*a(n-2)-8*a(n-3)+8*a(n-4) for n>6.
(End)
EXAMPLE
a(4) = 5 because 1122, 1123, 1132, 1213, 1133 are all Lyndon words on 3 letters with 2 ones.
MATHEMATICA
nn=30; Drop[CoefficientList[Series[(1/(1-2x)^2-1/(1-2x^2))/2, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 28 2013 *)
PROG
(PARI) Vec(x^3*(2-3*x)/((1-2*x)^2*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Oct 28 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mike Zabrocki, Nov 05 2006
STATUS
approved