OFFSET
1,3
COMMENTS
The number of binary strings sharing the same autocorrelations.
Appears to be row sums of A155092, beginning from a(2). - Mats Granvik, Jan 20 2009
The number of binary words of length n (beginning with 0) which do not start with an even palindrome (i.e. which are not of the form ss*t where s is a (nonempty) word, s* is its reverse, and t is any (possibly empty) word). - Mamuka Jibladze, Sep 30 2014
From Gus Wiseman, Mar 08 2021: (Start)
This sequence counts each of the following essentially equivalent things:
1. Sets of distinct positive integers with maximum n in which all adjacent elements have quotients > 1/2. For example, the a(1) = 1 through a(6) = 10 sets are:
{1} {2} {3} {4} {5} {6}
{2,3} {3,4} {3,5} {4,6}
{2,3,4} {4,5} {5,6}
{2,3,5} {3,4,6}
{3,4,5} {3,5,6}
{2,3,4,5} {4,5,6}
{2,3,4,6}
{2,3,5,6}
{3,4,5,6}
{2,3,4,5,6}
2. For n > 1, sets of distinct positive integers with maximum n - 1 whose first-differences are term-wise less than their decapitation (remove the maximum). For example, the set q = {2,4,5} has first-differences (2,1), which are not less than (2,4), so q is not counted under a(5). On the other hand, r = {2,3,5,6} has first-differences {1,2,1}, which are less than {2,3,5}, so r is counted under a(6).
3. Compositions of n where each part after the first is less than the sum of all preceding parts. For example, the a(1) = 1 through a(6) = 10 compositions are:
(1) (2) (3) (4) (5) (6)
(21) (31) (41) (51)
(211) (32) (42)
(311) (411)
(212) (321)
(2111) (312)
(3111)
(2121)
(2112)
(21111)
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..3324 (first 500 terms from T. D. Noe)
E. H. Rivals, Autocorrelation of Strings.
E. H. Rivals, S. Rahmann Combinatorics of Periods in Strings
E. H. Rivals, S. Rahmann, Combinatorics of Periods in Strings, Journal of Combinatorial Theory - Series A, Vol. 104(1) (2003), pp. 95-113.
FORMULA
a(2n) = 2*a(2n-1) - a(n) for n >= 1; a(2n+1) = 2*a(2n) for n >= 1.
a(n) = A342085(2^n). - Gus Wiseman, Mar 08 2021
MAPLE
a:= proc(n) option remember; `if`(n=0, 1/2,
2*a(n-1)-`if`(n::odd, 0, a(n/2)))
end:
seq(a(n), n=1..40); # Alois P. Heinz, Jun 24 2021
MATHEMATICA
a[1] = 1; a[n_] := a[n] = If[EvenQ[n], 2*a[n-1] - a[n/2], 2*a[n-1]]; Array[a, 40] (* Jean-François Alcover, Jul 17 2015 *)
Table[Length[Select[Subsets[Range[n]], MemberQ[#, n]&&Min@@Divide@@@Partition[#, 2, 1]>1/2&]], {n, 8}] (* Gus Wiseman, Mar 08 2021 *)
PROG
(PARI) a(n)=if(n<2, n>0, 2*a(n-1)-(1-n%2)*a(n\2))
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Torsten.Sillke(AT)uni-bielefeld.de
EXTENSIONS
More terms from James A. Sellers.
Additional comments from Michael Somos, Jun 09 2000
STATUS
approved