A053644 -id:A053644 - cp's OEIS frontend

A287730 The s-fusc function s(n) = a(n): a(1) = 0, a(2n) = A287729(n), a(2n+1) = A287729(n) + A287729(n+1).

0, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4

Offset: 1

I. V. Serov, May 30 2017

Define a sequence chf(n) of Christoffel words over an alphabet {-,+}:
  chf(1) = '-',
  chf(2*n+0) = negate(chf(n)),
  chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))).
Then the length of the chf(n) word is fusc(n) = A002487(n), the number of '-'-signs in the chf(n) word is c-fusc(n) = A287729(n) and the number of '+'-signs in the chf(n) word is the current sequence a(n) = s-fusc(n). See examples below.

			n         chf(n) A070939(n) A002487(n) A287729(n)    a(n)
                                fusc       c-fusc     s-fusc
1          '-'       1          1          1          0
2          '+'       2          1          0          1
3          '+-'      2          2          1          1
4          '-'       3          1          1          0
5          '--+'     3          3          2          1
6          '-+'      3          2          1          1
7          '-++'     3          3          1          2
8          '+'       4          1          0          1
9          '+++-'    4          4          1          3
10         '++-'     4          3          1          2
11         '++-+-'   4          5          2          3
12         '+-'      4          2          1          1
13         '+-+--'   4          5          3          2
14         '+--'     4          3          2          1
15         '+---'    4          4          3          1
16         '-'       5          1          1          0
17         '----+'   5          5          4          1

I. V. Serov (terms 1..1025) & Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to Stern's sequences

Cf. A002487, A007814, A070939, A037227, A287729, A053644.

Cf. mutual recurrence pair A000360, A284556 and also A213369.

from sympy.core.cache import cacheit
@cacheit
def c(n): return 1 if n==1 else s(n//2) if n%2==0 else s((n - 1)//2) + s((n + 1)//2)
@cacheit
def s(n): return 0 if n==1 else c(n//2) if n%2==0 else c((n - 1)//2) + c((n + 1)//2)
print([s(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 08 2017

(definec (A287730 n) (cond ((= 1 n) 0) ((even? n) (A287729 (/ n 2))) (else (+ (A287729 (/ (- n 1) 2)) (A287729 (/ (+ n 1) 2))))))
;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
;; Second version after the alternative formula given by the author:
(definec (A287730 n) (if (<= n 2) (- n 1) (- (* (A037227 (- n 1)) (A287730 (- n 1))) (A287730 (- n 2)) (* (if (= 1 (A002487 (- n 1))) 1 0) 2 (expt -1 (A070939 n)))))) ;; Antti Karttunen, Jun 01 2017

The mutual diatomic recurrence pair c(n) (A287729) and s(n) (this sequence) are defined by c(1)=1, s(1)=0, c(2n) = s(n), c(2n+1) = s(n)+s(n+1), s(2n) = c(n), s(2n+1) = c(n)+c(n+1).
a(n) + A287729(n) = A002487(n). [s-fusc(n) + c-fusc(n) = fusc(n).]
gcd(a(n), A287729(n)) = gcd(a(n), A002487(n)) = 1.
Let k(n) = A037227(n) = 1 + 2*A007814(n) = 1 + 2*floor(A002487(n-1)/A002487(n)) for n > 1.
Let d(n) = 2*A255738(n)*(-1)^A070939(n) = 2*(n==2^(A070939(n)-1)+1)*(-1)^A070939(n) = 2*(n==A053644(n)+1)*(-1)^A070939(n) = 2*(A002487(n-1)==1)*(-1)^A070939(n) for n > 1;
then a(n) = k(n-1)*a(n-1) - a(n-2) - d(n) for n > 2 with a(1) = 0, a(2) = 1.

A004756 Binary expansion starts 100.

Original entry on oeis.org

4, 8, 9, 16, 17, 18, 19, 32, 33, 34, 35, 36, 37, 38, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 1

N. J. A. Sloane

			18 in binary is 10010, so 18 is in sequence.

T. D. Noe, Table of n, a(n) for n = 1..1023
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A004754 (10), A004755 (11), A004757 (101), A004758 (110), A004759 (111).

Cf. A004760, A053644, A062050, A076877.

import Data.List (transpose)
a004756 n = a004756_list !! (n-1)
a004756_list = 4 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004756_list
-- Reinhard Zumkeller, Dec 04 2015

Select[Range[4, 153], Take[IntegerDigits[#, 2], 3] == {1, 0, 0} &] (* Michael De Vlieger, Aug 07 2016 *)

PARI
```
a(n)=n+3*2^floor(log(n)/log(2))
```

def A004756(n): return n+(3<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 3*[n==0].
a(n) = n + 3 * 2^floor(log_2(n)) = A004755(n) + A053644(n).
a(2^m+k) = 2^(m+2) + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 07 2016

Edited by Ralf Stephan, Oct 12 2003

A076877 a(n) = A020330(n) / n.

Original entry on oeis.org

3, 5, 5, 9, 9, 9, 9, 17, 17, 17, 17, 17, 17, 17, 17, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 129, 129, 129, 129, 129, 129

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

			12 -> '1100' -> '1100'1100' = '11001100' -> 204 = A020330(12): a(12) = A020330(12)/12 = 204/12 = 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.

Cf. A000523, A020330, A053644, A062383.

a[n_] := 1 + 2^Floor[Log2[n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 07 2021 *)

a(n) = 1 + 2^(1 + Log2(n)), with Log2 = A000523.
a(n) = 1 + 2*A053644(n).
a(n) = 1 + A062383(n).

A101402 a(0)=0, a(1)=1; for n>=2, let k = smallest power of 2 that is >= n, then a(n) = a(k/2) + a(n-1-k/2).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 27

Offset: 0

Odimar Fabeny, Jan 16 2005

Either a(n) = a(n-1) or a(n) = a(n-1) + 1. Proof: Suppose n is a power of 2, then a(n+1) = a(n) + a(0) = a(n). Otherwise let 2m be the largest power of 2 greater than n, so a(n) = a(m) + a(n-1-m) and a(n+1) = a(m) + a(n-m) and then proceed by induction. - Charles R Greathouse IV, Aug 27 2014
It appears that this sequence gives the partial sums of A164349. - Arie Groeneveld, Aug 27 2014
Each term other than zero appears at least twice. Suppose m is a power of 2, then a(2m) and a(4m) appear at least twice by my above comment. Otherwise suppose 3 <= k+2 <= 2m, then a(2m+k) = a(m) + a(m+k-1), a(2m+k+1) = a(m) + a(2m+k), and a(2m+k+2) = a(m) + a(m) + a(m+k+1), so a(2m+k+2) - a(2m+k) = a(m+k+1) - a(m+k-1). So if each term from a(m) to a(2m) appears at least twice then so will each term in a(2m) to a(4m). - Charles R Greathouse IV, Sep 10 2014
a(n) = Theta(n), see link. - Benoit Jubin, Sep 16 2014
The position of where n first appears: 0, 1, 4, 6, 10, 13, 15, 18, 21, 23, 27, 30, 32, 34, 37, 39, 43, 46, 48, 51, 54, 56, 60, 63, 66, 69, ... - Robert G. Wilson v, Sep 19 2014
The (10^k)-th term: 0, 3, 36, 355, 3549, 35494, 354942, ... - Robert G. Wilson v, Sep 19 2014

			a(2) = a(1) + a(0) = 1 = 1 + 0;
a(3) = a(2) + a(0) = 1 = 1 + 0;
a(4) = a(2) + a(1) = 2 = 1 + 1;
a(5) = a(4) + a(0) = 2 = 2 + 0;
a(6) = a(4) + a(1) = 3 = 2 + 1;
a(7) = a(4) + a(2) = 3 = 2 + 1;
a(8) = a(4) + a(3) = 3 = 2 + 1;
a(9) = a(8) + a(0) = 3 = 3 + 0; ...
The terms fall naturally into blocks of sizes 1,1,1,2,4,8,16,32,...:
0,
1,
1,
1, 2,
2, 3, 3, 3,
3, 4, 4, 4, 5, 5, 6, 6,
6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12,
12, 13, 13, 13, 14, 14, ...
Then the definition says that the k-th block is the final term of the previous block added to the sequence starting from the beginning (e.g., 34445566 = 3 + 01112233).
The final terms of the blocks, a(2^k), appear to be given by A164363. - _N. J. A. Sloane_, Aug 27 2014

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Benoit Jubin, Proof that A101402(n) = Theta(n)

Cf. A101403, A101404, A000045, A164363, A164369, A053644, A053645, A062383.

import Data.Function (on); import Data.List (genericIndex)
a101402 = genericIndex a101402_list
a101402_list = 0 : 1 : zipWith ((+) `on` a101402)
                       (tail a053644_list) a053645_list
-- Reinhard Zumkeller, Aug 27 2014

a[0] = 0; a[1] = 1; a[n_] := a[n] = Block[{p = 2^(Ceiling[Log[2, n]] - 1)}, a[p] + a[n - 1 - p]]; Table[ a@n, {n, 0, 100}] (* Robert G. Wilson v, Aug 17 2009 *)

a(n)=if(n<4, n>0, my(k=2^(log(n-.5)\log(2))); a(k) + a(n-1-k)) \\ Charles R Greathouse IV, Aug 25 2014

For n > 1: a(n) = a(A053644(n-1)) + a(A053645(n-1)). - Reinhard Zumkeller, Aug 27 2014

Offset corrected by R. J. Mathar, Aug 17 2009

More terms from Robert G. Wilson v, Aug 17 2009

A004759 Binary expansion starts 111.

Original entry on oeis.org

7, 14, 15, 28, 29, 30, 31, 56, 57, 58, 59, 60, 61, 62, 63, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244

Offset: 1

N. J. A. Sloane

This is the minimal recursive sequence such that a(1)=7, A007814(a(n))= A007814(n) and A010060(a(n))=A010060(n). - Vladimir Shevelev, Apr 23 2009

			30 in binary is 11110, so 30 is in sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..4095

Cf. A004754 (10), A004755 (11), A004756 (100), A004757 (101), A004758 (110).
Cf. A004760, A053644, A062050, A076877.
Cf. A007814, A010060, A103204, A159559, A159560, A159615, A159619, A159629, A159698. -Vladimir Shevelev, Apr 23 2009

import Data.List (transpose)
a004759 n = a004759_list !! (n-1)
a004759_list = 7 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004759_list
-- Reinhard Zumkeller, Dec 03 2015

w = {1, 1, 1}; Select[Range[5, 244], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 10 2016 *)
Sort[FromDigits[#,2]&/@(Flatten[Table[Join[{1,1,1},#]&/@Tuples[{1,0},n],{n,0,5}],1])] (* Harvey P. Dale, Sep 01 2016 *)

PARI
```
a(n)=n+6*2^floor(log(n)/log(2))
```

def A004759(n): return n+(3<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 6[n==0].
a(n) = n + 6 * 2^floor(log_2(n)) = A004758(n) + A053644(n).
a(n+1) = min{m > a(n): A007814(m) = A007814(n+1) and A010060(m) = A010060(n+1)}. a(2^k) - a(2^k-1) = A103204(k+2), k >= 1. - Vladimir Shevelev, Apr 23 2009
a(2^m+k) = 7*2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 3, 7, 1, 2, 2, 5, 3, 6, 7, 15, 1, 2, 2, 5, 2, 4, 5, 11, 3, 6, 6, 13, 7, 14, 15, 31, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 3, 6, 6, 13, 6, 12, 13, 27, 7, 14, 14, 29, 15, 30, 31, 63, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 2, 4, 4, 9, 4, 8, 9, 19, 5, 10, 10, 21, 11, 22, 23, 47, 3, 6, 6, 13, 6, 12, 13, 27, 6, 12, 12, 25, 13

Offset: 0

Antti Karttunen, Mar 24 2012

This is the number k such that the k-th composition in standard order is the reversed sequence of lengths of the maximal anti-runs of the binary indices of n. Here, the binary indices of n are row n of A048793, and the k-th composition in standard order is row k of A066099. For example, the binary indices of 100 are {3,6,7}, with maximal anti-runs ((3,6),(7)), with reversed lengths (1,2), which is the 6th composition in standard order, so a(100) = 6. - Gus Wiseman, Jul 27 2025

			102 in binary is 1100110, we rewrite it from the left so that first two 1's stay same ("11"), then "001" is rewritten to "0", the last 1 to "1", and we ignore the last 0, thus getting 1101, which is binary expansion of 13, thus a(102) = 13.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

This is an "inverse" of A071162, i.e. a(A071162(n)) = n for all n. Bisection: A209639. Used to construct permutation A209862.
Removing duplicates appears to give A358654.
Sorted positions of firsts appearances appear to be A247648+1.
A245563 lists run-lengths of binary indices (ranks A246029), reverse A245562.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A000120, A023758, A029931, A044813, A048793, A052499, A069010, A348366, A384878, A384879, A384893.
Cf. A384877, A385816, A385887, A385889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Reverse[Length/@Split[bpe[n],#2!=#1+1&]]],{n,0,100}] (* Gus Wiseman, Jul 25 2025 *)

import re
def a(n): return int(re.sub("0+1", "0", bin(n)[2:].rstrip("0")), 2) if n else 0
print([a(n) for n in range(109)])  # Michael S. Branicky, Jul 25 2025

(define (A209859 n) (let loop ((n n) (s 0) (i (A053644 n))) (cond ((zero? n) s) ((> i n) (if (> (/ i 2) n) (loop n s (/ i 2)) (loop (- n (/ i 2)) (* 2 s) (/ i 4)))) (else (loop (- n i) (+ (* 2 s) 1) (/ i 2))))))

a(n) = a(A000265(n)).

A348296 Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 4, 2, 4, 1, 2, 4, 8, 1, 8, 2, 8, 1, 2, 8, 4, 8, 1, 4, 8, 2, 4, 8, 1, 2, 4, 8, 16, 1, 16, 2, 16, 1, 2, 16, 4, 16, 1, 4, 16, 2, 4, 16, 1, 2, 4, 16, 8, 16, 1, 8, 16, 2, 8, 16, 1, 2, 8, 16, 4, 8, 16, 1, 4, 8, 16, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32

Offset: 1

Rémy Sigrist, Jul 18 2022

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------
   1  [1]
   2  [2]
   3  [1, 2]
   4  [4]
   5  [1, 4]
   6  [2, 4]
   7  [1, 2, 4]
   8  [8]
   9  [1, 8]
  10  [2, 8]
  11  [1, 2, 8]
  12  [4, 8]
  13  [1, 4, 8]
  14  [2, 4, 8]
  15  [1, 2, 4, 8]

Michael De Vlieger, Table of n, a(n) for n = 1..10870 (rows n = 1..2000, flattened)
Index entries for sequences related to binary expansion of n

Cf. A000079, A000120, A006519, A053644, A059867, A065620, A133457.

Array[DeleteCases[Union@ NumberExpand[#, 2], 0] &, 32] // Flatten (* Michael De Vlieger, Jul 19 2022 *)

row(n) = { my (r=vector(hammingweight(n))); for (k=1, #r, n -= r[k] = 2^valuation(n, 2)); return (r) }

T(n, k) = 2^A133457(n, k).
T(n, 1) = A006519(n).
T(n, A000120(n)) = A053644(n).
Sum_{k = 1..A000120(n)} T(n, k) = n.
Sum_{k = 1..A000120(n)} T(n, k) * (-1)^(k-1) = A065620(n).
Product_{k = 1..A000120(n)} T(n, k) = A059867(n).
T(2*n, k) = 2*T(n, k).

A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 05 2024

First differs from A335428 at n = 36. Differs from A050377, A344417 and A347437 at n = 1 and then at n = 36.

In the unique factorization of n in terms of distinct "Fermi-Dirac primes", n is represented as a product of prime powers (A246655) whose exponents are powers of 2 (A000079). a(n) is the maximal exponent of these prime powers (and not the maximal exponent of the exponents that are powers of 2). Thus, a(n) is a power of 2 for n >= 2.

			For n = 972 = 2^2 * 3^5, the unique factorization of 972 in terms of distinct "Fermi-Dirac primes" is 2^(2^1) * 3^(2^0) * 3^(2^2). Therefore, a(972) = 2^2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Fermi-Dirac prime.

Cf. A000079, A050376, A051903, A053644, A182979, A246655, A299090, A353897.

Cf. A050377, A335428, A344417, A347437.

a[n_] := 2^Floor[Log2[Max[FactorInteger[n][[;; , 2]]]]]; a[1] = 0; Array[a, 100]

a(n) = if(n > 1, 2^exponent(vecmax(factor(n)[, 2])), 0);

from sympy import factorint
def A368781(n): return 1<1 else 0 # Chai Wah Wu, Apr 11 2025

a(n) = A053644(A051903(n)).
a(n) = 2^(A299090(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2^(k-1) * (1 - 1/zeta(2^k))) = 1.56056154773294953123... .
a(n) = A051903(A353897(n)). - Amiram Eldar, May 07 2024

A088370 Triangle T(n,k), read by rows, where the n-th row is a binary arrangement of the numbers 1 through n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 5, 3, 2, 4, 1, 5, 3, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 14, 4, 12, 8

Offset: 1

Paul D. Hanna, Sep 28 2003

The n-th row differs from the prior row only by the presence of n. See A088371 for the positions in the n-th row that n is inserted.
From Clark Kimberling, Aug 02 2007: (Start)
At A131966, this sequence is cited as the fractal sequence of the Cantor set C.
Recall that C is the set of fractions in [0,1] whose base 3 representation consists solely of 0's and 2's.
Arrange these fractions as follows:
0
0, .2
0, .02, .2
0, .02, .2, .22
0, .002, .02, .2, .22, etc.
Replace each number x by its order of appearance, counting each distinct predecessor of x only once, getting
1;
1, 2;
1, 3, 2;
1, 3, 2, 4;
1, 5, 3, 2, 4;
Concatenate these to get the current sequence, which is a fractal sequence as defined in "Fractal sequences and interspersions".
One property of such a sequence is that it properly contains itself as a subsequence (infinitely many times). (End)
Row n contains one of A003407(n) non-averaging permutations of [n], i.e., a permutation of [n] without 3-term arithmetic progressions. - Alois P. Heinz, Dec 05 2017

			Row 5 is formed from row 3, {1,3,2} and row 2, {1,2}, like so:
{1,5,3, 2,4} = {1*2-1, 3*2-1, 2*2-1} | {1*2, 2*2}.
Triangle begins:
  1;
  1,  2;
  1,  3, 2;
  1,  3, 2,  4;
  1,  5, 3,  2,  4;
  1,  5, 3,  2,  6,  4;
  1,  5, 3,  7,  2,  6,  4;
  1,  5, 3,  7,  2,  6,  4,  8;
  1,  9, 5,  3,  7,  2,  6,  4,  8;
  1,  9, 5,  3,  7,  2, 10,  6,  4,  8;
  1,  9, 5,  3, 11,  7,  2, 10,  6,  4,  8;
  1,  9, 5,  3, 11,  7,  2, 10,  6,  4, 12,  8;
  1,  9, 5, 13,  3, 11,  7,  2, 10,  6,  4, 12,  8;
  1,  9, 5, 13,  3, 11,  7,  2, 10,  6, 14,  4, 12,  8;
  1,  9, 5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8;
  1,  9, 5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8, 16;
  1, 17, 9,  5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8, 16;
  ...

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Nonaveraging Sequence
Wikipedia, Arithmetic progression
Index entries related to non-averaging sequences

Cf. A003407, A088371, A309371.

Diagonal gives A053644. Cf. A049773. - Alois P. Heinz, Oct 28 2011

T:= proc(n) option remember;
      `if`(n=1, 1, [map(x-> 2*x-1, [T(n-iquo(n,2))])[],
                    map(x-> 2*x,   [T(  iquo(n,2))])[]][])
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2011

T[1] = {1}; T[n_] := T[n] = Join[q = Quotient[n, 2]; 2*T[n-q]-1, 2*T[q]]; Table[ T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

{T(n,k) = if(k==0, 1, if(k<=n\2, 2*T(n\2,k) - 1, 2*T((n-1)\2,k-1-n\2) ))}
for(n=0,20,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,n) = 2^(floor(log(n)/log(2))). Construction. The 2n-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n, after multiplying each term by 2. The (2n-1)-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n-1, after multiplying each term by 2.

Sum_{k=1..n} k * A088370(n,k) = A309371(n). - Alois P. Heinz, Jul 26 2019

A086099 a(n) = OR(k AND (n-k): 0<=k<=n), AND and OR bitwise.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 3, 0, 7, 6, 7, 4, 7, 6, 7, 0, 15, 14, 15, 12, 15, 14, 15, 8, 15, 14, 15, 12, 15, 14, 15, 0, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 16, 31, 30, 31, 28, 31, 30, 31, 24, 31, 30, 31, 28, 31, 30, 31, 0, 63, 62, 63, 60, 63, 62, 63, 56, 63, 62

Offset: 0

Reinhard Zumkeller, Jul 09 2003

a(2^n - 1) = 0, a(3*2^n - 1) = 2^n;

A086100(n) = A007088(a(n)).

			a(4) = (0 AND 4) OR (1 AND 3) OR (2 AND 2) OR (3 AND 1) OR (4 AND 0) -> (000 AND 100) OR (001 AND 011) OR (010 AND 010) OR (011 AND 001) OR (111 AND 000) = 000 OR 011 OR 010 OR 011 OR 000 = 011 -> a(4)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, AND
Eric Weisstein's World of Mathematics, OR
Reinhard Zumkeller, Logical Convolutions

Cf. A003817 (even bisection), A062383.
Cf. A086100 (in binary), A007088.
Cf. A053644, A006519.
Cf. A000004, A142149, A142150, A142151, A001477.

import Data.Bits ((.&.), (.|.))
a086099 n = foldl1 (.|.) $ zipWith (.&.) [0..] $ reverse [0..n] :: Integer
-- Reinhard Zumkeller, Jun 04 2012

a[n_] := BitOr @@ Table[BitAnd[k, n - k], {k, 0, n}]; Table[a[n], {n, 0, 73}] (* Jean-François Alcover, Jun 19 2012 *)

PARI
```
a(n) = n++; 1<Kevin Ryde, Apr 11 2023
```

a(2*n) = 2*2^floor(log_2(n)) - 1 = A003817(n).
a(2*n+1) = 2*a(n).
a(n) = A053644(n+1) - A006519(n+1). - Ridouane Oudra, Apr 09 2023

cp's OEIS Frontend

A287730 The s-fusc function s(n) = a(n): a(1) = 0, a(2n) = A287729(n), a(2n+1) = A287729(n) + A287729(n+1).

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A004756 Binary expansion starts 100.

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A076877 a(n) = A020330(n) / n.

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A101402 a(0)=0, a(1)=1; for n>=2, let k = smallest power of 2 that is >= n, then a(n) = a(k/2) + a(n-1-k/2).

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A004759 Binary expansion starts 111.

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A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

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A348296 Irregular table T(n, k), n > 0, k = 1..A000120(n), read by rows; the n-th contains, in ascending order, the distinct powers of 2 summing to n.