|
|
A080653
|
|
a(1) = 2; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) such that the condition "a(a(n)) is always even" is satisfied.
|
|
7
|
|
|
2, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 42, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 95, 96, 97
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
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
|
|
|
FORMULA
|
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
|
|
|
STATUS
|
approved
|
|
|
|