cp's OEIS Frontend

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.

%I A080653 #28 Dec 14 2022 10:07:32
%S A080653 2,4,5,6,8,10,11,12,13,14,16,18,20,22,23,24,25,26,27,28,29,30,32,34,
%T A080653 36,38,40,42,44,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,64,
%U A080653 66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,95,96,97
%N 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.
%C A080653 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.
%C A080653 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.
%C A080653 Lower density 2/3, upper density 3/4. - _Charles R Greathouse IV_, Dec 14 2022
%D A080653 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
%H A080653 Yifan Xie, <a href="/A080653/b080653.txt">Table of n, a(n) for n = 1..10000</a>
%H A080653 Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A080653 Benoit Cloitre, N. J. A. Sloane and Matthew J. Vandermast, <a href="https://arxiv.org/abs/math/0305308">Numerical analogues of Aronson's sequence</a> (math.NT/0305308)
%F A080653 a(a(n)) = 2n + 2. - _Yifan Xie_, Jul 14 2022
%F A080653 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
%t A080653 (* 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 *)
%Y A080653 Equals A007378 - 2.
%Y A080653 A007378, A079905, A080637, A080653 are all essentially the same sequence.
%Y A080653 Cf. A169956, A169957.
%K A080653 easy,nonn,nice
%O A080653 1,1
%A A080653 _Matthew Vandermast_, Mar 01 2003