A079945 -id:A079945 - cp's OEIS frontend

A079944 A run of 2^n 0's followed by a run of 2^n 1's, for n=0, 1, 2, ...

0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Feb 21 2003

With offset 2, this is the second bit in the binary expansion of n. - Franklin T. Adams-Watters, Feb 13 2009

a(n) = A173920(n+2,2); in the sequence of nonnegative integers (cf. A001477) substitute all n by 2^floor(n/2) occurrences of (n mod 2). - Reinhard Zumkeller, Mar 04 2010

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. See Example 1.34.

R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions

Cf. A086694, A079882, A079945, A173922, A173923.

a079944 n = a079944_list !! n
a079944_list =  f [0,1] where f (x:xs) = x : f (xs ++ [x,x])
-- Reinhard Zumkeller, Oct 14 2010, Mar 28 2011

Table[IntegerDigits[n + 2, 2][[2]], {n, 0, 100}] (* Jean-François Alcover, Jul 26 2019 *)

a(n)=binary(n+2)[2] \\ Charles R Greathouse IV, Nov 07 2016

a(n) = floor(log[2](4*(n+2)/3)) - floor(log[2](n+2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003
For n >= 2, a(n-2)=1+floor(log[2](n/3))-floor(log[2](n/2)) - Benoit Cloitre, Mar 03 2003
G.f.: 1/x^2/(1-x) * (1/x + sum(k>=0, x^(3*2^k)-x^2^(k+1))). - Ralf Stephan, Jun 04 2003
a(n) = A000035(A004526(A030101(n+2))). - Reinhard Zumkeller, Mar 04 2010

A079882 A run of 2^n 1's followed by a run of 2^n 2's, for n=0, 1, 2, ...

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Feb 21 2003

In the sequence of nonnegative integers (cf. A001477) substitute all n by 2^floor(n/2) occurrences of (1 + n mod 2); a(n)=A173920(n+2,3) for n>0. [From Reinhard Zumkeller, Mar 04 2010]

Robert Israel, Table of n, a(n) for n = 0..10000

Partial sums give A079945. Equals 1 + A079944. Cf. A080584.

First differences of A080637.

f1 := n->[seq(1,i=1..2^n)]; f2 := n->[seq(2,i=1..2^n)]; s := []; for i from 0 to 10 do s := [op(s), op(f1(i)), op(f2(i))]; od: s;

Table[{PadRight[{},2^n,1],PadRight[{},2^n,2]},{n,0,5}]//Flatten (* Harvey P. Dale, Jul 22 2016 *)

a(n) = floor(log[2](8*(n+2)/3)) - floor(log[2](n+2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

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

A080596 a(1)=1; for n >= 2, a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+3.

Original entry on oeis.org

1, 4, 5, 7, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

N. J. A. Sloane, Feb 23 2003

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

Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000. Apart from initial terms, same as A079945.

a(1) = 1; then a(6*2^k-3+j) = 8*2^k-3+3j/2+|j|/2 for k >= 0, -2^(k+1) <= j < 2^(k+1).

A080702 a(1)=3; for n>1, a(n) = smallest number > a(n-1) such that the condition "if n is in the sequence then a(n) is even" is satisfied.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Benoit Cloitre, Mar 01 2003

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

Equals A079945(n+1) - 1.

lista(nn) = {v = vector(nn); v[1] = 3; prev = v[1]; for (n=2, nn, new = prev+1; if (vecsearch(vecsort(v,,8), n) && (new % 2), new ++); v[n] = new; prev = new;); v;} \\ Michel Marcus, Feb 16 2015

{a(a(n))} = {2i : i >= 3}.

More terms from Matthew Vandermast, Mar 05 2003

cp's OEIS Frontend

A079944 A run of 2^n 0's followed by a run of 2^n 1's, for n=0, 1, 2, ...

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A079882 A run of 2^n 1's followed by a run of 2^n 2's, for n=0, 1, 2, ...

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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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A080596 a(1)=1; for n >= 2, a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+3.

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A080702 a(1)=3; for n>1, a(n) = smallest number > a(n-1) such that the condition "if n is in the sequence then a(n) is even" is satisfied.

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