login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A124720 Number of ternary Lyndon words of length n with exactly two 1's. 6
2, 5, 16, 38, 96, 220, 512, 1144, 2560, 5616, 12288, 26592, 57344, 122816, 262144, 556928, 1179648, 2490112, 5242880, 11009536, 23068672, 48233472, 100663296, 209713152, 436207616, 905965568, 1879048192, 3892305920, 8053063680, 16642981888, 34359738368 (list; graph; refs; listen; history; text; internal format)
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 binay 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

Cf. A051168, A027376, A124721, A124722, A124723.

Sequence in context: A082085 A179992 A054971 * A188947 A076958 A163825

Adjacent sequences:  A124717 A124718 A124719 * A124721 A124722 A124723

KEYWORD

nonn,easy

AUTHOR

Mike Zabrocki, Nov 05 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 7 00:47 EDT 2020. Contains 333291 sequences. (Running on oeis4.)