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.

Showing 1-5 of 5 results.

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,
    ...

Links

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).

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

Links

Crossrefs

See A330940 for the minimum variant.
Cf. A002001, A024036, A191755, A193020.

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).

A330940 a(n) is the least 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, 10, 15, 36, 43, 54, 63, 136, 147, 170, 175, 204, 219, 238, 255, 528, 547, 586, 591, 660, 683, 694, 703, 792, 819, 858, 879, 924, 955, 990, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2358, 2367, 2600, 2643, 2730, 2735, 2764, 2779, 2798, 2815, 3120, 3171
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 representation of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     3       1         11
   2    10      10       1010
   3    15      11       1111
   4    36     100     100100
   5    43     101     101011
   6    54     110     110110
   7    63     111     111111
   8   136    1000   10001000
   9   147    1001   10010011
  10   170    1010   10101010
  11   175    1011   10101111

Links

Crossrefs

See A330941 for the maximum variant.
Cf. A007582, A024036, A191755, A193020.

Programs

  • PARI
    See Links section.

Formula

a(2^k) = 2^k*(1+2^(k+1)) = A007582(k+1) for any k >= 0.
a(2^k-1) = 4^k-1 = A024036(k) for any k >= 0.
a(n) <= A330941(n).

A361398 An infiltration of two words, say x and y, is a shuffle of x and y optionally followed by replacements of pairs of consecutive equal symbols, say two d's, one of which comes from x and the other from y, by a single d (that cannot be part of another replacement); a(n) is the number of distinct infiltrations of the word given by the binary representation of n with itself.

Original entry on oeis.org

1, 2, 5, 3, 9, 12, 9, 4, 14, 28, 30, 21, 19, 21, 14, 5, 20, 53, 68, 60, 55, 74, 68, 32, 34, 60, 55, 36, 34, 32, 20, 6, 27, 89, 126, 134, 120, 181, 196, 108, 88, 181, 183, 136, 151, 164, 126, 45, 55, 134, 151, 129, 107, 136, 120, 54, 69, 108, 88, 54, 55, 45, 27
Offset: 0

Views

Author

Rémy Sigrist, Mar 10 2023

Keywords

Comments

Leading zeros in binary expansions are ignored.
See A191755 for the definition of a shuffle.

Examples

			For n = 2:
- the binary expansion of 2 is "10",
- we have essentially the following infiltrations:
         x        10   10    1 0   10     1 0
         y        10   1 0    10     10    1 0
                  --   ---   ---   ----   ----
    infiltration  10   100   110   1010   1100
- so a(2) = 5.

Links

Crossrefs

Cf. A000096, A191755, A193020.

Programs

  • PARI
    See Links section.

Formula

a(n) >= A193020(n).
a(2^k - 1) = k + 1 for any k >= 0.
a(2^k) = A000096(k + 1) for any k >= 0.

A303476 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is the number of distinct shuffles of the words corresponding to the binary representations of n and of k.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 3, 3, 3, 3, 2, 3, 1, 3, 2, 2, 4, 6, 6, 4, 2, 1, 3, 3, 3, 3, 3, 1, 4, 4, 3, 7, 7, 3, 4, 4, 3, 4, 1, 6, 4, 6, 1, 4, 3, 3, 6, 10, 10, 6, 6, 10, 10, 6, 3, 2, 4, 6, 4, 4, 3, 4, 4, 6, 4, 2, 3, 6, 6, 9, 11, 4, 4, 11, 9, 6, 6, 3, 2, 4, 3, 7, 8, 10, 1
Offset: 1

Views

Author

Rémy Sigrist, Apr 24 2018

Keywords

Comments

A shuffle of two words is formed by interspersing their characters into a new word, keeping the characters of each word in order. Leading zeros are ignored.

Examples

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12
  ---+------------------------------------------------
    1|   1   2   1   3   2   2   1   4   3   3   2   3
    2|   2   2   3   3   4   3   4   4   6   4   6   4
    3|   1   3   1   6   3   3   1  10   6   6   3   6
    4|   3   3   6   3   7   6  10   4   9   7  13   6
    5|   2   4   3   7   4   6   4  11   8   8   6  10
    6|   2   3   3   6   6   3   4  10  12   7   9   6
    7|   1   4   1  10   4   4   1  20  10  10   4  10
    8|   4   4  10   4  11  10  20   4  13  11  24  10
    9|   3   6   6   9   8  12  10  13   9  15  14  18
   10|   3   4   6   7   8   7  10  11  15   8  14  11

Links

Crossrefs

Cf. A008687, A193020.

Formula

T(n, k) = T(k, n).
T(n, n) = A193020(n).
Apparently T(n, 1) = A008687(n + 1).
T(2^i, 2^j) = 1 + max(i, j) for any i >=0 and j >= 0.
T(n, k) = 1 iff n = 2^i - 1 and k = 2^j - 1 for some i > 0 and j > 0.
T(2^i, 2^j - 1) = binomial(i + j, j) for any i >= 0 and j > 0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.