3,1

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

a(n) is the number of distinct positive rationals smaller than one that contain n bits when written in binary: a(4) = 5 because the rationals 1/100, 1/101, 1/110, 1/111, 10/11 are all smaller than unity. - Johannes M.V.A. Koelman, Jan 17 2026

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.

Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

G.f.: x^3*(2-3*x)/((1-2*x^2)*(1-2*x)^2) = (x^2/(1-2*x)^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)

E.g.f.: (3 + exp(2*x)*(2*x - 1) - 2*cosh(sqrt(2)*x))/8. - Stefano Spezia, Jan 23 2026

a(4) = 5 because 1122, 1123, 1132, 1213, 1133 are all Lyndon words on 3 letters with 2 ones.

nn=30; Drop[CoefficientList[Series[(1/(1-2x)^2-1/(1-2x^2))/2, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 28 2013 *)

(PARI) Vec(x^3*(2-3*x)/((1-2*x)^2*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Oct 28 2016

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

Sequence in context: A082085 A179992 A054971 * A188947 A076958 A163825

Adjacent sequences: A124717 A124718 A124719 * A124721 A124722 A124723

nonn,easy

Mike Zabrocki, Nov 05 2006

approved