A283188 A periodic sequence of 8-bit binary numbers for single-bit multi-frequency generation.

255, 127, 191, 31, 207, 71, 163, 33, 240, 80, 152, 24, 236, 108, 174, 6, 215, 87, 179, 51, 235, 73, 137, 9, 252, 116, 180, 20, 198, 70, 170, 42, 251, 91, 155, 17, 229, 101, 165, 5, 220, 92, 186, 58, 234, 66, 130, 2, 247, 117, 189, 29, 205, 77, 169, 33, 242, 82, 146, 18, 238, 110, 174, 12, 221

Offset

0,1

Comments

If the updating frequency of numbers is f, then the most significant bit (MSB) changes with frequency f, while the second MSB changes with f/2, the third MSB changes with f/3, fourth MSB with f/4, and so on till the least significant bit (LSB), which changes with f/8.

The length of the cyclic period is 1680 = lcm(2,4,6,8,10,12,14,16). Only 192 numbers are used out of the 256 possible 8-bit binary numbers. Possible number of repetitions within a cycle are 6, 8, 9, and 12.

For a fast continuous production of the sequence, it could be more efficient to implement a cyclic lookup table rather than calculating the numbers each time.

A similar sequence can be obtained from A272614 (excluding the first 8 terms) by selecting the 8 most significant bits after having removed the MSB.

Links

Table of n, a(n) for n=0..64.

Formula

a(n) = Sum_{k = 1..8} (floor((n - k)/k) mod 2)*2^(8 - k).

Example

Binary expansions of the first 16 terms:
  [1, 1, 1, 1, 1, 1, 1, 1]
  [0, 1, 1, 1, 1, 1, 1, 1]
  [1, 0, 1, 1, 1, 1, 1, 1]
  [0, 0, 0, 1, 1, 1, 1, 1]
  [1, 1, 0, 0, 1, 1, 1, 1]
  [0, 1, 0, 0, 0, 1, 1, 1]
  [1, 0, 1, 0, 0, 0, 1, 1]
  [0, 0, 1, 0, 0, 0, 0, 1]
  [1, 1, 1, 1, 0, 0, 0, 0]
  [0, 1, 0, 1, 0, 0, 0, 0]
  [1, 0, 0, 1, 1, 0, 0, 0]
  [0, 0, 0, 1, 1, 0, 0, 0]
  [1, 1, 1, 0, 1, 1, 0, 0]
  [0, 1, 1, 0, 1, 1, 0, 0]
  [1, 0, 1, 0, 1, 1, 1, 0]
  [0, 0, 0, 0, 0, 1, 1, 0]

Mathematica

a[n_] := Sum[Floor@Mod[(n - k)/k, 2]*2^(8 - k), {k, 1, 8}];
Table[a[n], {n, 0, 64}]

Prog

(PARI) a(n) = sum(k=1, 8, ((floor((n - k) / k)) % 2)*2^(8 - k)); \\ Indranil Ghosh, Mar 03 2017
(Python)
def A283188(n):
    s=0
    for k in range(1, 9):
        s+=(((n-k)//k)%2) * 2**(8-k)
    return s # Indranil Ghosh, Mar 03 2017

Crossrefs

Cf. A272614.

Sequence in context: A251860 A145715 A145587 * A334353 A028526 A132828

Adjacent sequences: A283185 A283186 A283187 * A283189 A283190 A283191

Keyword

nonn,easy

Author

Andres Cicuttin, Mar 02 2017

Status

approved