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

Author

Darío Clavijo, Jul 14 2024

Keywords

• nonn
• base
• easy

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.

Examples

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

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.

Crossrefs

Cf. A030101, A115637, A179888.

Programs

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 *)

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

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.