A080653 -id:A080653 - cp's OEIS frontend

A007378 a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n.

3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 97, 98, 99, 100, 101, 102, 103

Offset: 2

Colin Mallows

This is the unique monotonic sequence {a(n)}, n>=2, satisfying a(a(n)) = 2n.
May also be defined by: a(n), n=2,3,4,..., is smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is an even number >= 4". - N. J. A. Sloane, Feb 23 2003
A monotone sequence satisfying a^(k+1)(n) = mn is unique if m=2, k >= 0 or if (k,m) = (1,3). See A088720. - Colin Mallows, Oct 16 2003
Numbers (greater than 2) whose binary representation starts with "11" or ends with "0". - Franklin T. Adams-Watters, Jan 17 2006
Lower density 2/3, upper density 3/4. - Charles R Greathouse IV, Dec 14 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
J.-P. Allouche, N. Rampersad and J. Shallit, On integer sequences whose first iterates are linear, Aequationes Math. 69 (2005), 114-127.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit 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.
Jeffrey Shallit, k-regular Sequences
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences of the a(a(n)) = 2n family

Cf. A003605. Equals A080653 + 2.
This sequence, A079905, A080637 and A080653 are all essentially the same.
Cf. A088720, A042965.

a := proc(n) option remember; if n < 4 then n+1 else a(iquo(n,2)) + a(iquo(n+1,2)) fi end:
seq(a(n), n = 2..70); # Peter Bala, Aug 03 2022

max = 70; f[x_] := -x/(1-x) + x/(1-x)^2*(2 + Sum[ x^(2^k + 2^(k+1)) - x^2^(k+1) , {k, 0, Ceiling[Log[2, max]]}]); Drop[ CoefficientList[ Series[f[x], {x, 0, max + 1}], x], 2](* Jean-François Alcover, May 16 2012, from g.f. *)
a[2]=3; a[3]=4; a[n_?OddQ] := a[n] = a[(n-1)/2+1] + a[(n-1)/2]; a[n_?EvenQ] := a[n] = 2a[n/2]; Table[a[n], {n, 2, 71}] (* Jean-François Alcover, Jun 26 2012, after Vladeta Jovovic *)

a(n) = my(s=logint(n,2)-1); if(bittest(n,s), n<<1 - 2<Kevin Ryde, Aug 08 2022

from functools import cache
@cache
def a(n): return n+1 if n < 4 else a(n//2) + a((n+1)//2)
print([a(n) for n in range(2, 72)]) # Michael S. Branicky, Aug 04 2022

a(2^i + j) = 3*2^(i-1) + j, 0<=j<2^(i-1); a(3*2^(i-1) + j) = 2^(i+1) + 2*j, 0<=j<2^(i-1).
a(3*2^k + j) = 4*2^k + 3j/2 + |j|/2, k>=0, -2^k <= j < 2^k. - N. J. A. Sloane, Feb 23 2003
a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n). a(n) = n+A060973(n). - Vladeta Jovovic, Mar 01 2003
G.f.: -x/(1-x) + x/(1-x)^2 * (2 + sum(k>=0, t^2(t-1), t=x^2^k)). - Ralf Stephan, Sep 12 2003

More terms from Matthew Vandermast and Vladeta Jovovic, Mar 01 2003

A080637 a(n) is the smallest positive integer which is consistent with the sequence being monotonically increasing and satisfying a(1)=2, a(a(n)) = 2n+1 for n > 1.

Original entry on oeis.org

2, 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 and Benoit Cloitre, Feb 28 2003

Sequence is the unique monotonic sequence satisfying a(a(n)) = 2n+1.

Except for the first term, numbers (greater than 2) whose binary representation starts with 11 or ends with 1. - Yifan Xie, May 26 2022

			From _Yifan Xie_, May 02 2022: (Start)
a(8) = 12 because 2*2^2 <= 8 < 3*2^2, hence a(8) = 8 + 2^2 = 12;
a(13) = 19 because 3*2^2 <= 13 < 4*2^2, hence a(13) = 2*(13 - 2^2) + 1 = 19. (End)

Yifan Xie, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit 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

Except for first term, same as A079905. Cf. A079000.
A007378, A079905, A080637, A080653 are all essentially the same sequence.
Equals A007378(n+1)-1. First differences give A079882.
Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3: this sequence (k=2), A003605 (k=3), A353651 (k=4), A353652 (k=5), A353653 (k=6).

t := []; for k from 0 to 6 do for j from -2^k to 2^k-1 do t := [op(t), 4*2^k - 1 + 3*j/2 + abs(j)/2]; od: od: t;

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_] := b[n+1]-1;
a /@ Range[70] (* Jean-François Alcover, Oct 31 2019 *)

a(3*2^k - 1 + j) = 4*2^k - 1 + 3*j/2 + |j|/2 for k >= 0, -2^k <= j < 2^k.
a(2n+1) = 2*a(n) + 1, a(2n) = a(n) + a(n-1) + 1.
From Yifan Xie, May 02 2022: (Start)
For n in the range 2*2^i <= n < 3*2^i, for i >= 0:
  a(n) = n + 2^i.
  a(n) = 1 + a(n-1).
Otherwise, for n in the range 3*2^i <= n < 4*2^i, for i >= 0:
  a(n) = 2*(n - 2^i) + 1.
  a(n) = 2 + a(n-1). (End)

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.

Original entry on oeis.org

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

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

A169957 a(1) = 1, then a(n) is the smallest number such that a(a(n)) = 2n+1.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 10, 13, 17, 15, 14, 19, 25, 23, 18, 21, 33, 27, 22, 35, 41, 31, 26, 29, 49, 39, 30, 51, 57, 47, 34, 37, 65, 43, 38, 67, 73, 55, 42, 45, 81, 71, 46, 83, 89, 63, 50, 53, 97, 59, 54, 99, 105, 79, 58, 61, 113, 103, 62, 115

Offset: 1

Eric Angelini, Aug 02 2010

Original name was "Similar to A079905, but without the 'monotonically increasing sequence' condition."

Yifan Xie, Table of n, a(n) for n = 1..10000 (first 499 terms from Charlie Neder)

Cf. A080653, A079905, A169956.

a(1) = 1, a(2) = 3, a(4k) = 4k + 2, all other terms follow from the criterion defining the sequence. - Charlie Neder, Dec 21 2018

a(25)-a(61) from Charlie Neder, Dec 21 2018

Name changed by Charlie Neder, Dec 22 2018

A115836 Self-describing sequence. The n-th integer of the sequence indicates how many integers of the sequence are strictly < 2n.

Original entry on oeis.org

1, 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, 98, 99, 100, 101

Offset: 1

Eric Angelini, Feb 01 2006

Terms computed by Gilles Sadowski. In building step by step the sequence one has sometimes the choice for an integer. If so take the smallest available one.

{a(n)} is the lexicographically earliest monotonic sequence of positive integers satisfying a(a(n)+1) = 2*n. - Yifan Xie, Jun 25 2024

			a(7)=10 because there are 10 integers in the sequence which are strictly < 14 (they are 1,2,4,5,6,8,10,11,12,13)

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.

Cf. A007378, A080653.

a(n) = A007378(n+1) - 2. - Benoit Cloitre, May 22 2008

cp's OEIS Frontend

A007378 a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n.

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A080637 a(n) is the smallest positive integer which is consistent with the sequence being monotonically increasing and satisfying a(1)=2, a(a(n)) = 2n+1 for n > 1.

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

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A169957 a(1) = 1, then a(n) is the smallest number such that a(a(n)) = 2n+1.

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Links

Crossrefs

Formula

Extensions

A115836 Self-describing sequence. The n-th integer of the sequence indicates how many integers of the sequence are strictly < 2n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula