OFFSET
1,1
COMMENTS
Also defined by: a(n) = smallest positive number > a(n-1) such that the condition "n is in sequence if and only if a(n) is odd" is false (cf. A079000); that is, the condition "either n is not in the sequence and a(n) is odd or n is in the sequence and a(n) is even" is satisfied.
If prefixed with a(0) = 0, can be defined by: a(n) = smallest nonnegative number > a(n-1) such that the condition "n is in sequence only if a(n) is even" is satisfied.
Lower density 2/3, upper density 3/4. - Charles R Greathouse IV, Dec 14 2022
REFERENCES
Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585
LINKS
Yifan Xie, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
FORMULA
a(a(n)) = 2n + 2. - Yifan Xie, Jul 14 2022
a(n+1) - a(n) is in {1, 2}. In particular, n < a(n) <= 2n. More is true: lim inf a(n)/n = 4/3 and lim sup a(n)/n = 3/2. - Charles R Greathouse IV, Dec 14 2022
MATHEMATICA
(* b = A007378 *) b[n_] := b[n] = Which[n == 2, 3, n == 3, 4, EvenQ[n], 2 b[n/2], True, b[(n-1)/2+1]+b[(n-1)/2]]; a[1] = 2; a[n_] := b[n+2]-2; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 05 2016 *)
CROSSREFS
KEYWORD
easy,nonn,nice
AUTHOR
Matthew Vandermast, Mar 01 2003
STATUS
approved