A062880 -id:A062880 - cp's OEIS frontend

A062033 Binary expansion of n does not contain 1-bits at even positions. Integers whose base 4 representation consists of only 0's and 2s.

0, 10, 1000, 1010, 100000, 100010, 101000, 101010, 10000000, 10000010, 10001000, 10001010, 10100000, 10100010, 10101000, 10101010, 1000000000, 1000000010, 1000001000, 1000001010, 1000100000, 1000100010, 1000101000, 1000101010, 1010000000, 1010000010, 1010001000

Offset: 0

Antti Karttunen, Jun 26 2001

Decimal representation is given in A062880.

Cf. A063010.

def A062033(n): return int(bin(int(bin(n)[2:],4))[2:])*10 # Chai Wah Wu, Aug 21 2023

More terms from Chai Wah Wu, Aug 22 2023

A342217 The n-th and a(n)-th points of the Hilbert's Hamiltonian walk (A059252, A059253) are symmetrical with respect to the line X=Y.

Original entry on oeis.org

0, 3, 2, 1, 14, 15, 12, 13, 8, 11, 10, 9, 6, 7, 4, 5, 58, 57, 56, 59, 60, 63, 62, 61, 50, 49, 48, 51, 52, 55, 54, 53, 32, 35, 34, 33, 46, 47, 44, 45, 40, 43, 42, 41, 38, 39, 36, 37, 26, 25, 24, 27, 28, 31, 30, 29, 18, 17, 16, 19, 20, 23, 22, 21, 234, 235, 232

Offset: 0

Rémy Sigrist, Mar 05 2021

nonn

In other words, a(n) is the unique k such that A059252(n) = A059253(k) and A059253(n) = A059252(k).

This sequence is a self-inverse permutation of the nonnegative integers.

			The Hilbert's Hamiltonian walk (A059252, A059253) begins as follows:
     +     +-----+-----+
     |15   |12    11   |10
     |     |           |
     +-----+     +-----+
      14    13   |8     9
                 |
     +-----+     +-----+
     |1    |2     7    |6
     |     |           |
     +     +-----+-----+
      0     3     4     5
- so a(0) = 0,
     a(1) = 3,
     a(2) = 2,
     a(4) = 14,
     a(5) = 15,
     a(7) = 13,
     a(8) = 8,
     a(9) = 11,
     a(10) = 10.

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Rémy Sigrist, PARI program for A342217
Index entries for sequences that are permutations of the nonnegative integers

See A342218 and A342224 for similar sequences.

Cf. A059252, A059253, A062880.

PARI
```
See Links section.
```

a(n) = n iff n belongs to A062880.

a(n) < 16^k for any n < 16^k.

A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jun 13 2021

This sequence is a variant of A022290; here we consider Fibonacci numbers with negative indices (A039834), there Fibonacci numbers with positive indices (A000045).

After the initial 0, the sequence alternates runs of positive terms and runs of negative terms, the k-th run having 2^(k-1) terms.

			For n = 3:
- 3 = 2^1 + 2^0,
- so a(3) = A039834(2+1) + A039834(2+0) = 2 - 1 = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Cf. A000045, A000695, A020989, A022290, A039834, A062880, A063694, A063695, A072197, A080675, A136412, A345291, A345292.

a(n) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=fibonacci(-2-e)); v }

a(n) = A022290(A063695(n)) - A022290(A063694(n)).
a(n) = A022290(n) iff n belongs to A062880.
a(n) = -A022290(n) iff n belongs to A000695.
a(n) = 0 iff n = 0.
a(n) = 1 iff n belongs to A072197.
a(n) = 2 iff n belongs to A080675.
a(n) = -1 iff n belongs to A020989.
a(n) = -2 iff n belongs to A136412.

A147568 a(n) = 2*A000695(n)+3.

Original entry on oeis.org

3, 5, 11, 13, 35, 37, 43, 45, 131, 133, 139, 141, 163, 165, 171, 173, 515, 517, 523, 525, 547, 549, 555, 557, 643, 645, 651, 653, 675, 677, 683, 685, 2051, 2053, 2059, 2061, 2083, 2085, 2091, 2093, 2179, 2181, 2187, 2189, 2211, 2213, 2219, 2221, 2563, 2565, 2571

Offset: 0

Vladimir Shevelev, Nov 07 2008

nonn

Every odd number m>=9 is a unique sum of the form a(k)+2a(l); moreover this sequence is the unique one with such property. In connection with A103151, note that there is no subsequence T of primes such that every odd number m>=9 is expressible as a unique sum of the form m=p+2q, where p and q are in T. One can prove that if one replaces 9 by any integer x_o>9, the statement remains true (see the Shevelev link).

Vladimir Shevelev, On Unique Additive Representations of Positive Integers and Some Close Problems, arXiv:0811.0290 [math.NT], 2008.

Cf. A000695, A062880, A103151.

(* b = A000695 *) b[n_] := If[n==0, 0, If[EvenQ[n], 4 b[n/2] , b[n-1]+1]];
a[n_] := 2 b[n] + 3; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 14 2018 *)

a000695(n) = fromdigits(binary(n), 4);
a(n) = 2*a000695(n)+3; \\ Michel Marcus, Dec 13 2018

More terms from Michel Marcus, Dec 13 2018

A341120 a(n) is the X-coordinate of the n-th point of the space filling curve C defined in Comments section; A341121 gives Y-coordinates.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 2, 3, 4, 4, 3, 3, 3, 4, 5, 5, 5, 4, 4, 5, 6, 6, 6, 7, 8, 8, 7, 7, 7, 8, 8, 7, 6, 6, 5, 5, 5, 6, 5, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 8, 8, 9, 10, 10, 10, 11, 12, 12, 11, 11, 11, 12, 12, 13

Offset: 0

Kevin Ryde and Rémy Sigrist, Feb 05 2021

nonn

We define the family {C_k, k >= 0}, as follows:
- C_0 corresponds to the points (0, 0), (0, 1), (1, 1), (2, 1) and (2, 0), in that order:
       +---+---+
       |       |
       +       +
      O
- for any k >= 0, C_{k+1} is obtained by arranging 4 copies of C_k as follows:
                           + . . . + . . . +
                           .   B   .   B   .
       + . . . +           .       .       .
       .   B   .           .A     C.A     C.
       .       .    -->    + . . . + . . . +
       .A     C.           .C      .      A.
       + . . . +           .      B.B      .
      O                    .A      .      C.
                           + . . . + . . . +
                          O
- the space filling curve C is the limit of C_{2*k} as k tends to infinity.
The even bisection of the curve M defined in A341018 is similar to C and vice versa.
The third quadrisection of C is similar to the Hilbert Hamiltonian walk H = A059252, A059253.
H is the number of points in the middle of each unit square in Hilbert's subdivisions, whereas here points are at the starting corner of each unit square. This start is either the bottom left or top right corner depending on how many 180-degree rotations have been applied. These rotations are digit 3's of n written in base 4, hence the formula below adding A283316.

			Points n and their locations X=a(n), Y=A341121(n) begin as follows. n=7 and n=9 are both at X=3,Y=2, and n=11,n=31 both at X=3,Y=4.
      |       |
    4 | 16---17   12--11,31
      |  |         |    |
    3 | 15---14---13   10
      |                 |
    2 |            8---7,9
      |                 |
    1 |  1----2----3    6
      |  |         |    |
  Y=0 |  0         4----5
      +--------------------
       X=0    1    2    3

Rémy Sigrist, Table of n, a(n) for n = 0..16384
F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982. See section 4.8.
David Hilbert, Ueber die stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen, volume 38, number 3, 1891, pages 459-460. Also EUDML (link to GDZ).
Rémy Sigrist, Illustration of C_6
Rémy Sigrist, Illustration of the connection of C with the Hilbert Hamiltonian walk
Rémy Sigrist, Illustration of the connection of C with the space filling curve M defined in A341018
Rémy Sigrist, PARI program for A341120
Index entries for sequences related to coordinates of 2D curves

Cf. A341121 (Y coordinate), A059285 (projection Y-X), A062880 (n on X=Y diagonal).

Cf, A059252, A341018.

PARI
```
See Links section.
```

a(n) = A341121(n) - A059285(n).
a(n) = A341121(n) iff n belongs to A062880.
a(2*n) = A341018(n).
a(4*n) = 2*A341121(n).
a(16*n) = 4*a(n).
a(n) = A059252(n) + A283316(n+1).
A059253(n) = (a(4*n+2)-1)/2.

A045110 Numbers whose base-4 representation contains no 1's and exactly one 3.

Original entry on oeis.org

3, 11, 12, 14, 35, 43, 44, 46, 48, 50, 56, 58, 131, 139, 140, 142, 163, 171, 172, 174, 176, 178, 184, 186, 192, 194, 200, 202, 224, 226, 232, 234, 515, 523, 524, 526, 547, 555, 556, 558, 560, 562, 568, 570, 643, 651, 652, 654, 675

Offset: 1

Clark Kimberling

From Robert Israel, Apr 10 2018: (Start)
If k is in the sequence, then so are 4*k and 4*k+2.
All even terms of the sequence are obtained in this way.
The odd terms of the sequence are 4*k+3 for k in A062880. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062880, A007090.

f:= proc(i,x,d) local j,X, nX;
     X:= convert(x,base,2); nX:= nops(X);
     3*4^i+add(X[j]*2*4^`if`(j<=i,j-1,j),j=1..nX)
end proc:
sort([seq(seq(seq(f(i,x,d),x=`if`(i=d-1,0,2^(d-2))..2^(d-1)-1),i=0..d-1),d=0..6)]); # Robert Israel, Apr 10 2018

Select[Range[700],DigitCount[#,4,1]==0&&DigitCount[#,4,3]==1&] (* Harvey P. Dale, Jul 23 2018 *)

A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

Original entry on oeis.org

2, 8, 1, 10, 3, 5, 32, 4, 7, 21, 34, 6, 13, 23, 85, 40, 9, 15, 29, 87, 341, 42, 11, 17, 31, 93, 343, 1365, 128, 12, 19, 53, 95, 349, 1367, 5461, 130, 14, 20, 55, 117, 351, 1373, 5463, 21845, 136, 16, 22, 61, 119, 373, 1375, 5469, 21847, 87381, 138, 18, 25, 63

Offset: 1

Clark Kimberling, Sep 12 2009

For n>=0, row n is the ordered sequence of positive integers m such that the number of even powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For odd powers, see A165275.
For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A062880, (Row 1)=A158705, (Column 1)=A002450, also possibly (Column 2)=A163832.

			Northwest corner:
2....8...10...32...34...40...42...129
1....3....4....6....9...11...12...14
5....7...13...15...17...19...20...22
21..23...29...31...53...55...61...63
Examples:
40 = 32 + 8 = 2^5 + 2^3, so that 40 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 2.

Cf. A139351, A139352, A165275, A165276, A165277, A165278, A165279.

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)

More terms from Amiram Eldar, Feb 04 2020

A331961 a(n) is the greatest square number k such that n AND k = k (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 0, 1, 4, 4, 4, 4, 0, 9, 0, 9, 4, 9, 4, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 16, 25, 16, 25, 16, 25, 0, 1, 0, 1, 36, 36, 36, 36, 0, 9, 0, 9, 36, 36, 36, 36, 16, 49, 16, 49, 36, 49, 36, 49, 16, 49, 16, 49, 36, 49, 36, 49, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Rémy Sigrist, Feb 02 2020

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     0      10          0
   3     1      11          1
   4     4     100        100
   5     4     101        100
   6     4     110        100
   7     4     111        100
   8     0    1000          0
   9     9    1001       1001
  10     0    1010          0
  11     9    1011       1001
  12     4    1100        100
  13     9    1101       1001
  14     4    1110        100
  15     9    1111       1001
  16    16   10000      10000

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A062880, A330270.

a(n) = forstep (m=sqrtint(n), 0, -1, if (bitand(n, m^2)==m^2, return (m^2)))

from math import isqrt
def A331961(n): return next(m for m in (k**2 for k in range(isqrt(n),-1,-1)) if n&m==m) # Chai Wah Wu, Aug 22 2023

a(n) = 0 iff n belongs to A062880.

a(n^2) = n^2.

A176237 Binary expansion of n contains at least one 1-bit at even position and one 1-bit at odd position.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 13, 14, 15, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93

Offset: 1

Rémy Sigrist, Apr 12 2010

Also numbers neither in A000695 nor in A062880.

			The binary expansion of 6 is "110", and has one 1-bit at (odd) position 1 and one 1-bit at (even) position 2; thus 6 appears in the sequence.

Cf. A000695, A062880.

ok[n_] := With[{p = Position[IntegerDigits[n, 2], 1]}, AnyTrue[p, OddQ[#[[1]]]&] && AnyTrue[p, EvenQ[#[[1]]]&]]; Select[Range[100], ok] (* Jean-François Alcover, Mar 31 2017 *)

A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i2^(i-1) and Sum_{i > 0} c_i2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i2^(i-1) with d_i = (Sum_{k divides i} b_kc_{i/k}) mod 2 for any i > 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 32, 9, 32, 5, 0, 0, 6, 34, 36, 36, 34, 6, 0, 0, 7, 40, 39, 256, 39, 40, 7, 0, 0, 8, 42, 46, 260, 260, 46, 42, 8, 0, 0, 9, 128, 45, 288, 257, 288, 45, 128, 9, 0, 0, 10, 130, 136, 292, 294, 294, 292, 136, 130, 10, 0

Offset: 0

Rémy Sigrist, Apr 30 2024

The set of nonnegative integers equipped with A form a commutative monoid.

			Array A(n, k) begins:
  n\k | 0   1    2    3     4     5     6     7      8      9     10
  ----+-------------------------------------------------------------
    0 | 0   0    0    0     0     0     0     0      0      0      0
    1 | 0   1    2    3     4     5     6     7      8      9     10
    2 | 0   2    8   10    32    34    40    42    128    130    136
    3 | 0   3   10    9    36    39    46    45    136    139    130
    4 | 0   4   32   36   256   260   288   292   2048   2052   2080
    5 | 0   5   34   39   260   257   294   291   2056   2061   2090
    6 | 0   6   40   46   288   294   264   270   2176   2182   2216
    7 | 0   7   42   45   292   291   270   265   2184   2191   2210
    8 | 0   8  128  136  2048  2056  2176  2184  32768  32776  32896
    9 | 0   9  130  139  2052  2061  2182  2191  32776  32769  32906
   10 | 0  10  136  130  2080  2090  2216  2210  32896  32906  32776

Cf. A000120, A062880, A048720, A341288.

bits(n) = { my (b=vector(hammingweight(n))); for (k=1, #b, n-=2^b[k]=valuation(n,2)); return (b); }
A(n, k) = { my (bn = bits(2*n), bk = bits(2*k), v = 0, e); for (i = 1, #bn, for (j = 1, #bk, e = bn[i] * bk[j] - 1; v = bitxor(v, 2^e););); return (v); }

A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k).
A(m XOR n, k) = A(m, k) XOR A(n, k) (where XOR denotes the bitwise XOR operator).
A000120(A(n, 2^k)) = A000120(n).
A(n, 0) = 0.
A(n, 1) = n.
A(n, 2) = A062880(n).

cp's OEIS Frontend

A062033 Binary expansion of n does not contain 1-bits at even positions. Integers whose base 4 representation consists of only 0's and 2s.

Views

Author

Keywords

Crossrefs

Programs

Python

Extensions

A342217 The n-th and a(n)-th points of the Hilbert's Hamiltonian walk (A059252, A059253) are symmetrical with respect to the line X=Y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A345290 a(n) is obtained by replacing 2^k in binary expansion of n with Fibonacci(-k-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A147568 a(n) = 2*A000695(n)+3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A341120 a(n) is the X-coordinate of the n-th point of the space filling curve C defined in Comments section; A341121 gives Y-coordinates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A045110 Numbers whose base-4 representation contains no 1's and exactly one 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A331961 a(n) is the greatest square number k such that n AND k = k (where AND denotes the bitwise AND operator).

Views

Author

Keywords

Examples

Links

Crossrefs

A370049 Square array A(n, k), n, k >= 0, read by antidiagonals; for any n and k >= 0 with respective binary expansions Sum_{i > 0} b_i2^(i-1) and Sum_{i > 0} c_i2^(i-1), the binary expansion of A(n, k) is Sum_{i > 0} d_i2^(i-1) with d_i = (Sum_{k divides i} b_kc_{i/k}) mod 2 for any i > 0.