A169957 -id:A169957 - cp's OEIS frontend

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.

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, 37, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

N. J. A. Sloane, Feb 21 2003

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

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

Yifan Xie, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Hsien-Kuei Hwang, S. Janson, and T.-H. 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.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, 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.
Index entries for sequences of the a(a(n)) = 2n family

See A080637 for a nicer version. Cf. A079000.
Equals A007378(n+1)-1, n>1.
A007378, A079905, A080637, A080653 are all essentially the same sequence.
Cf. A169956, A169957.
Union of A079946 and A005408 (the odd numbers).

b[n_] := b[n] = If[n<4, n+1, If[OddQ[n], b[(n-1)/2+1]+b[(n-1)/2], 2b[n/2]]];
a[n_] := If[n == 1, 1, b[n+1] - 1];
a /@ Range[70] (* Jean-François Alcover, Oct 18 2019, after Vladeta Jovovic in A007378 *)

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.

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

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003

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.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Mar 01 2003

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

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

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)

Equals A007378 - 2.
A007378, A079905, A080637, A080653 are all essentially the same sequence.
Cf. A169956, A169957.

(* 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 *)

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

A169956 Lexicographically earliest sequence with positive integers satisfying a(a(n)) = 2*n+2.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 9, 12, 16, 14, 13, 18, 24, 22, 17, 20, 32, 26, 21, 34, 40, 30, 25, 28, 48, 38, 29, 50, 56, 46, 33, 36, 64, 42, 37, 66, 72, 54, 41, 44, 80, 70, 45, 82, 88, 62, 49, 52, 96, 58, 53, 98, 104, 78, 57, 60, 112, 102, 61, 114, 120, 94, 65, 68, 128, 74

Offset: 1

Eric Angelini, Aug 02 2010

Previous name was: Similar to A080653, but without the "monotonically increasing sequence" condition.

Sequence contains all the even numbers and odd numbers k such that k mod 4 = 1. - Yifan Xie, Jul 05 2022

			For n=3, a(3) must satisfy a(a(3)) = 2*3+2 = 8. If a(3) = 0, we get 8 = a(a(3)) = a(0) = 1, so a(3) > 0. Using the same method twice we get a(3) > 2. If a(3) = 3, hence 3 = a(3) = a(a(3)) = 8, so a(3) > 3. If a(3) = 4, using a(2) = 4 we get 8 = a(a(3)) = a(4) = 2*2+2 = 6, so a(3) > 4. If a(3) = 5, there are no conflicts using the definition, so a(3) = 5. - _Yifan Xie_, Jul 05 2022

Yifan Xie, Table of n, a(n) for n = 1..10000

Cf. A080653, A079905, A169957.

From Yifan Xie, Jul 05 2022: (Start)
a(A169957(n)-1) = 2*n for n > 1.
a(4*n-1) = 4*n+1 for n >= 1.
a(4*n+1) = 8*n for n >= 1. (End)

New name and more terms from Yifan Xie, Jul 05 2022

a(23) corrected by Yifan Xie, Jul 05 2022

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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.

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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.

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A169956 Lexicographically earliest sequence with positive integers satisfying a(a(n)) = 2*n+2.

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