A374625 In the binary expansion of n, expand bits 1 -> 01 and 0 -> 10.

2, 1, 6, 5, 26, 25, 22, 21, 106, 105, 102, 101, 90, 89, 86, 85, 426, 425, 422, 421, 410, 409, 406, 405, 362, 361, 358, 357, 346, 345, 342, 341, 1706, 1705, 1702, 1701, 1690, 1689, 1686, 1685, 1642, 1641, 1638, 1637, 1626, 1625, 1622, 1621, 1450, 1449, 1446

Offset

0,1

Comments

This is essentialy the Manchester encoding (defined by convention in IEEE802.3 opposed to the original version where the mappings are inverted published by G. E. Thomas in 1949 see ref) which transforms each bit by representing 1 as 01 and 0 as 10, ensuring a transition occurs in the middle of each bit period for synchronization.

References

Andrew S. Tanenbaum, Computer Networks (4th ed.), Prentice Hall, 2002, Pages 274-275, ISBN 0-13-066102-3.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003, equation 2.4 with a(n) = a_n for case alpha=4, c=2, d=1 and n >= 1.

Wikipedia, Manchester code.

Formula

a(n) = a(n+1) + 1 for n even.

a(n) = A030101(A179888(A030101(n))) for n odd.

a(2*n) = 4*a(n) + 2 for n>=1.

a(2*n+1) = 4*a(n) + 1 for n>=1.

Example

a(5) = 25 because:
  5 is 101 and:
   1 |  0 |  1
  01 | 10 | 01
  And: 11001 is 25.

Maple

a:= n-> 2-(n mod 2)+`if`(n<2, 0, 4*a(iquo(n, 2))):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 15 2024

Mathematica

A374625[n_] := FromDigits[2 - IntegerDigits[n, 2], 4];
Array[A374625, 100, 0] (* Paolo Xausa, Jul 16 2024 *)

Prog

(Python)
def a(n):
  s=''
  for b in bin(n)[2:]:
    s += '01'*(b=='1') + '10'*(b=='0')
  return int(s, 2)
print([a(n) for n in range(0, 51)])
(Python)
def a(n):
  d = {'0': '10', '1': '01'}
  return int(''.join(map(d.get, bin(n)[2:])), 2)
print([a(n) for n in range(0, 51)]) # Jason Yuen, Jul 15 2024
(Python)
a = lambda n: int(''.join("101"[b=='1':(b=='1')+2] for b in bin(n)[2:]), 2)
# Peter Luschny, Jul 15 2024
(Python)
def A374625(n): return ((1<<(n.bit_length()<<1)+1)-2)//3-int(bin(n)[2:], 4) if n else 2 # Chai Wah Wu, Jul 16 2024

Crossrefs

Cf. A030101, A115637, A179888.

Sequence in context: A095132 A028940 A376980 * A218853 A048998 A213615

Adjacent sequences: A374622 A374623 A374624 * A374626 A374627 A374628

Keyword

nonn,base,easy

Author

Darío Clavijo, Jul 14 2024

Status

approved