A354780 -id:A354780 - cp's OEIS frontend

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

1, 3, 4, 12, 3, 19, 34, 64, 76, 136, 256, 768, 17, 1041, 50, 2080, 4096, 12288, 68, 16452, 200, 32896, 65536, 196608, 768, 262912, 524800, 1048576, 1049601, 2098176, 18, 4194322, 2096, 8390656, 16777216, 50331648, 12288, 67121152, 134225920, 268435456, 268451844, 536887296, 72, 1073741896, 32960, 2147516416, 4294967296, 12884901888

Offset: 1

N. J. A. Sloane, Jul 05 2022

			See A354780.

N. J. A. Sloane, Table of n, a(n) for n = 1..2400

Cf. A354169, A354680, A354757, A354758, A354767, A354773, A354774, A354778, A355150, A354780.

A354169 a(0) = 0, a(1) = 1, a(2) = 2; for k >= 2, given a(k), the sequence is extended by adjoining two terms: a(2k-1) = smallest m >= 0 not among a(0) .. a(k) such that {m, a(k), a(k+1), ..., a(2k-2)} are pairwise disjoint in binary, and a(2k) = smallest m >= 0 not among a(0) .. a(k) such that {m, a(k), ..., a(2k-1)} are pairwise disjoint in binary.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 12, 128, 256, 512, 17, 1024, 34, 2048, 4096, 8192, 68, 16384, 136, 32768, 65536, 131072, 768, 262144, 524288, 1048576, 1025, 2097152, 18, 4194304, 2080, 8388608, 16777216, 33554432, 12288, 67108864, 134217728, 268435456, 16388

Offset: 0

N. J. A. Sloane, Jun 05 2022

The paper by De Vlieger et al. (2022) calls this the "binary two-up sequence".
"Pairwise disjoint in binary" means no common 1-bits in their binary representations.
This is a set-theory analog of A090252. It bears the same relation to A090252 as A252867 does to A098550, A353708 to A121216, A353712 to A347113, etc.
A consequence of the definition, and also an equivalent definition, is that this is the lexicographically earliest infinite sequence of distinct nonnegative numbers with the property that the binary representation of a(n) is disjoint from (has no common 1's with) the binary representations of the following n terms.
An equivalent definition is that a(n) is the smallest nonnegative number that is disjoint (in its binary representation) from each of the previous floor(n/2) terms.
For the subsequence 0, 3, 12, 17, 34, ... of the terms that are not powers of 2 see A354680 and A354798.
All terms are the sum of at most two powers of 2 (see De Vlieger et al., 2022). - N. J. A. Sloane, Aug 29 2022

			After a(2) = 2 = 10_2, a(3) must equal ?0?_2, and the smallest such number we have not seen is a(3) = 100_2 = 4, and a(4) must equal ?00?_2, and the smallest such number we have not seen is a(4) = 1000_2 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..4941
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
Rémy Sigrist, PARI program
N. J. A. Sloane, Construction of terms a(0) through a(18)
N. J. A. Sloane, Table showing first 190 terms (pdf file of scan of large table)
N. J. A. Sloane, Left-hand portion of rows 97-135 of preceding table, showing columns 67-96, rotated counter-clockwise by 90 degrees. The red line in this table should be aligned with the red line between rows 135 and 136 in the preceding table. [This portion of the table was too wide to fit through the scanner.]
N. J. A. Sloane, Blog post about the Two-Up sequence, June 13 2022. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 16.

A355889 is a more efficient way to present this sequence.
Cf. A000120, A090252, A098550, A121216, A252867, A347113, A353708, A353712, A354680, A354767, A354798, A355150.
See also A354773, A354774, A354775, A354780, A354781, A354783.

nn = 42; c[] = r = 0; m = 1; Array[Set[{a[#], c[#]}, {#, #}] &, 2, 0]; Do[k = SelectFirst[Union@ Map[Total, Rest@ Subsets[2^Reverse[Length[#] - Position[#, 1][[All, 1]]] &@ IntegerDigits[2^(r + 2) - m - 1, 2]]], c[#] == 0 &]; Set[{a[n], c[k]}, {k, n}]; m += a[n]; If[And[IntegerQ[#], # > 0], m -= a[#]] &[n/2]; If[And[EvenQ[k], PrimePowerQ[k], k > 2^r], r++], {n, 2, nn}]; Table[a[n], {n, 0, nn}] (* _Michael De Vlieger, Jul 13 2022 *)

PARI
```
See Links section.
```

from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354169_gen(): # generator of terms
    aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
    yield from (0,1,2)
    while True:
        for k in count(1):
            m, j, j2, r, s = 0, 0, 1, b, k
            while r > 0:
                r, q = divmod(r,2)
                if not q:
                    s, y = divmod(s,2)
                    m += y*j2
                j += 1
                j2 *= 2
            if s > 0:
                m += s*2**b.bit_length()
            if m not in aset:
                yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354169_list = list(islice(A354169_gen(),40)) # Chai Wah Wu, Jun 06 2022

More terms from Rémy Sigrist, Jun 06 2022

A354757 a(n) = Sum_{k = ceiling(n/2)..n-1} A354169(k).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 15, 27, 59, 115, 127, 252, 508, 1004, 1021, 2013, 2047, 4031, 8127, 16307, 16375, 32631, 32767, 65279, 130815, 261375, 262143, 524270, 1048558, 2096110, 2097135, 4194253, 4194271, 8386527, 8388607, 16773119, 33550335, 67096575, 67108863

Offset: 0

Rémy Sigrist, Jun 06 2022

The 1's in the binary expansion of a(n) are forbidden in that of A354169(n). In other words, a(n) AND A354169(n) = 0 (where AND denotes the bitwise AND operator).

			a(5) = A354169(3) + A354169(4) = 4 + 8 = 12.
a(7) = A354169(4) + A354169(5) + A354169(6) = 8 + 3 + 16 = 27.

N. J. A. Sloane, Table of n, a(n) for n = 0..4800
Rémy Sigrist, Binary plot of a(n) (blue pixels) and A354169(n) (red pixels) for n=0..1000
Rémy Sigrist, PARI program

Cf. A354169, A354758.

A354780 is a bisection.

PARI
```
See Links section.
```

from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354757_gen(): # generator of terms
    aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
    yield from (0,0,1)
    while True:
        for k in count(1):
            m, j, j2, r, s = 0, 0, 1, b, k
            while r > 0:
                r, q = divmod(r,2)
                if not q:
                    s, y = divmod(s,2)
                    m += y*j2
                j += 1
                j2 *= 2
            if s > 0:
                m += s*2**b.bit_length()
            if m not in aset:
                yield sum(aqueue)
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354757_list = list(islice(A354757_gen(),40)) # Chai Wah Wu, Jun 06 2022

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 4, 4, 12, 0, 3, 3, 19, 2, 34, 0, 64, 64, 76, 8, 136, 0, 256, 256, 768, 0, 17, 17, 1041, 16, 50, 32, 2080, 0, 4096, 4096, 12288, 0, 68, 68, 16452, 64, 200, 128, 32896, 0, 65536, 65536, 196608, 0, 768, 768, 262912, 512, 524800, 0, 1048576, 1048576, 1049601, 1024, 2098176, 0, 18, 18, 4194322, 16, 2096, 2048, 8390656, 0, 16777216

Offset: 1

N. J. A. Sloane, Jul 08 2022

Has the same relation to A354757 as A354781 does to A354780.

The offset is 1, to avoid having to define a(0).

			A354757(5) = 12 = 1100_2, so a(5) = 11_2 = 3.
A354757(6) = 15 = 1111_2, so a(6) = 0.
A354757(7) = 27 = 11011_2, so a(7) = 100_2 = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..4800

Cf. A354169, A354757, A354780, A354781.

See A354793 for Hamming weight of a(n).

Added comment and examples. - N. J. A. Sloane, Aug 02 2022

cp's OEIS Frontend

A354781 If the binary expansion of A354780(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Views

Author

Keywords

Examples

Links

Crossrefs

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A354757 a(n) = Sum_{k = ceiling(n/2)..n-1} A354169(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A354783 If the binary expansion of A354757(n) is 1 d_1 d_2 ... d_k, then the binary expansion of a(n) is c_1 c_2 ... c_k, where c_i = 1 - d_i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions