A330941 -id:A330941 - cp's OEIS frontend

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.

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

Rémy Sigrist, Jan 04 2020

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

			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

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Logarithmic scatterplot of the first difference of the first 2^13 terms
Rémy Sigrist, PARI program for A330940
Index entries for sequences related to binary expansion of n

See A330941 for the maximum variant.

Cf. A007582, A024036, A191755, A193020.

PARI
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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).

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

Rémy Sigrist, Dec 05 2022

See A358892 for the distinct values.

n and T(n, k) have the same parity.

			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,
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..4768 (rows for n = 0..127 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

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

PARI
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T(n, 1) = A330940(n).

T(n, A193020(n)) = A330941(n).

A361401 Irregular table T(n, k), n >= 0, k = 1..A361398(n); the n-th row lists the numbers whose binary expansion is a self-infiltration of that of n.

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 10, 12, 3, 7, 15, 4, 8, 12, 16, 20, 24, 36, 40, 48, 5, 9, 11, 13, 19, 21, 25, 27, 43, 45, 51, 53, 6, 12, 14, 26, 28, 30, 54, 58, 60, 7, 15, 31, 63, 8, 16, 24, 32, 40, 48, 64, 72, 80, 96, 136, 144, 160, 192

Offset: 0

Rémy Sigrist, Mar 10 2023

See A361398 for the definition of an infiltration (a self-infiltration is an infiltration a of word with itself).

The terms of the n-th row of A358893 appear in the n-th row of the present table (they correspond to terms with twice as many binary digits as n).

			Table T(n, k) begins:
  n  n-th row
  -  ---------------------------------------------------------
  0  0
  1  1, 3
  2  2, 4, 6, 10, 12
  3  3, 7, 15
  4  4, 8, 12, 16, 20, 24, 36, 40, 48
  5  5, 9, 11, 13, 19, 21, 25, 27, 43, 45, 51, 53
  6  6, 12, 14, 26, 28, 30, 54, 58, 60
  7  7, 15, 31, 63
  8  8, 16, 24, 32, 40, 48, 64, 72, 80, 96, 136, 144, 160, 192

Rémy Sigrist, Table of n, a(n) for n = 0..10102 (rows for n = 0..82 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A330941, A358893, A361398.

PARI
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See Links section.
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T(n, 1) = 1.

T(n, A361398(n)) = A330941(n).

cp's OEIS Frontend

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.

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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.

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A361401 Irregular table T(n, k), n >= 0, k = 1..A361398(n); the n-th row lists the numbers whose binary expansion is a self-infiltration of that of n.

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