cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A330941 a(n) is the greatest value whose binary representation can be obtained by interleaving (or shuffling) two copies of the binary representation of n.

Original entry on oeis.org

0, 3, 12, 15, 48, 53, 60, 63, 192, 201, 212, 219, 240, 245, 252, 255, 768, 785, 804, 819, 848, 853, 876, 887, 960, 969, 980, 987, 1008, 1013, 1020, 1023, 3072, 3105, 3140, 3171, 3216, 3237, 3276, 3303, 3392, 3401, 3412, 3435, 3504, 3509, 3548, 3567, 3840, 3857
Offset: 0

Views

Author

Rémy Sigrist, Jan 04 2020

Keywords

Comments

The binary representation of all positive terms are square binary words (see A191755).

Examples

			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     3       1          11
   2    12      10        1100
   3    15      11        1111
   4    48     100      110000
   5    53     101      110101
   6    60     110      111100
   7    63     111      111111
   8   192    1000    11000000
   9   201    1001    11001001
  10   212    1010    11010100
  11   219    1011    11011011
  12   240    1100    11110000
		

Crossrefs

See A330940 for the minimum variant.

Programs

  • PARI
    See Links section.

Formula

a(2^k) = 3*4^k = A002001(k+1) for any k >= 0.
a(2^k-1) = 4^k-1 = A024036(k) for any k >= 0.
a(n) >= A330940(n).

A358893 Irregular triangle T(n, k), n >= 0, k = 1..A193020(n), read by rows: the n-th row lists the numbers obtained by self-shuffling the binary expansion of n.

Original entry on oeis.org

0, 3, 10, 12, 15, 36, 40, 48, 43, 45, 51, 53, 54, 58, 60, 63, 136, 144, 160, 192, 147, 149, 153, 163, 165, 169, 195, 197, 201, 170, 172, 178, 180, 202, 204, 210, 212, 175, 183, 187, 207, 215, 219, 204, 212, 216, 228, 232, 240, 219, 221, 235, 237, 243, 245
Offset: 0

Views

Author

Rémy Sigrist, Dec 05 2022

Keywords

Comments

See A358892 for the distinct values.
n and T(n, k) have the same parity.

Examples

			Triangle T begins (in decimal):
    n   n-th row
    --  --------
     0  0,
     1  3,
     2  10, 12,
     3  15,
     4  36, 40, 48,
     5  43, 45, 51, 53,
     6  54, 58, 60,
     7  63,
     8  136, 144, 160, 192,
     9  147, 149, 153, 163, 165, 169, 195, 197, 201,
     ...
Triangle T begins (in binary):
    n     n-th row
    ----  --------
       0  0,
       1  11,
      10  1010, 1100,
      11  1111,
     100  100100, 101000, 110000,
     101  101011, 101101, 110011, 110101,
     110  110110, 111010, 111100,
     111  111111,
    1000  10001000, 10010000, 10100000, 11000000,
    ...
		

Crossrefs

Cf. A193020 (row lengths), A330940, A330941, A358892.

Programs

  • PARI
    See Links section.

Formula

T(n, 1) = A330940(n).
T(n, A193020(n)) = A330941(n).
Showing 1-2 of 2 results.