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A079905 a(1)=1; then a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+1 for n>1.

%I A079905 #29 Jul 15 2022 04:08:31
%S A079905 1,3,5,6,7,9,11,12,13,14,15,17,19,21,23,24,25,26,27,28,29,30,31,33,35,
%T A079905 37,39,41,43,45,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,65,
%U A079905 67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,96,97,98,99,100,101,102
%N A079905 a(1)=1; then a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+1 for n>1.
%C A079905 Alternate definition: a(n) is taken to be smallest positive integer greater than a(n-1) such that the condition "a(a(n)) is always odd" can be satisfied. - _Matthew Vandermast_, Mar 03 2003
%C A079905 Also: a(n)=smallest positive integer > a(n-1) such that the condition "n is in the sequence if and only if a(n) is even" is false; 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. - _Matthew Vandermast_, Mar 05 2003
%H A079905 Yifan Xie, <a href="/A079905/b079905.txt">Table of n, a(n) for n = 1..10000</a>
%H A079905 B. Cloitre, N. J. A. Sloane and M. 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 A079905 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, <a href="https://arxiv.org/abs/math/0305308">Numerical analogues of Aronson's sequence</a>, arXiv:math/0305308 [math.NT], 2003.
%H A079905 Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, Preprint 2016.
%H A079905 Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
%H A079905 <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%F A079905 a(1)=1, a(2)=3, then a(3*2^k - 1 + j) = 4*2^k - 1 + 3j/2 + |j|/2 for k >= 1, -2^k <= j < 2^k.
%F A079905 a(n) = 1+A079945(n-1)-A079944(n-1) for n>1, a(1)=1. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
%t A079905 b[n_] := b[n] = If[n<4, n+1, If[OddQ[n], b[(n-1)/2+1]+b[(n-1)/2], 2b[n/2]]];
%t A079905 a[n_] := If[n == 1, 1, b[n+1] - 1];
%t A079905 a /@ Range[70] (* _Jean-François Alcover_, Oct 18 2019, after _Vladeta Jovovic_ in A007378 *)
%Y A079905 See A080637 for a nicer version. Cf. A079000.
%Y A079905 Equals A007378(n+1)-1, n>1.
%Y A079905 A007378, A079905, A080637, A080653 are all essentially the same sequence.
%Y A079905 Cf. A169956, A169957.
%Y A079905 Union of A079946 and A005408 (the odd numbers).
%K A079905 nonn,easy
%O A079905 1,2
%A A079905 _N. J. A. Sloane_, Feb 21 2003
%E A079905 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003