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

%I A374625 #51 Jul 17 2024 04:55:24
%S A374625 2,1,6,5,26,25,22,21,106,105,102,101,90,89,86,85,426,425,422,421,410,
%T A374625 409,406,405,362,361,358,357,346,345,342,341,1706,1705,1702,1701,1690,
%U A374625 1689,1686,1685,1642,1641,1638,1637,1626,1625,1622,1621,1450,1449,1446
%N A374625 In the binary expansion of n, expand bits 1 -> 01 and 0 -> 10.
%C A374625 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.
%D A374625 Andrew S. Tanenbaum, Computer Networks (4th ed.), Prentice Hall, 2002, Pages 274-275, ISBN 0-13-066102-3.
%H A374625 Paolo Xausa, <a href="/A374625/b374625.txt">Table of n, a(n) for n = 0..10000</a>
%H A374625 Ralf Stephan, <a href="https://arxiv.org/abs/math/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>, 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.
%H A374625 Wikipedia, <a href="https://en.wikipedia.org/wiki/Manchester_code">Manchester code</a>.
%F A374625 a(n) = a(n+1) + 1 for n even.
%F A374625 a(n) = A030101(A179888(A030101(n))) for n odd.
%F A374625 a(2*n) = 4*a(n) + 2 for n>=1.
%F A374625 a(2*n+1) = 4*a(n) + 1 for n>=1.
%e A374625 a(5) = 25 because:
%e A374625   5 is 101 and:
%e A374625    1 |  0 |  1
%e A374625   01 | 10 | 01
%e A374625   And: 11001 is 25.
%p A374625 a:= n-> 2-(n mod 2)+`if`(n<2, 0, 4*a(iquo(n, 2))):
%p A374625 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 15 2024
%t A374625 A374625[n_] := FromDigits[2 - IntegerDigits[n, 2], 4];
%t A374625 Array[A374625, 100, 0] (* _Paolo Xausa_, Jul 16 2024 *)
%o A374625 (Python)
%o A374625 def a(n):
%o A374625   s=''
%o A374625   for b in bin(n)[2:]:
%o A374625     s += '01'*(b=='1') + '10'*(b=='0')
%o A374625   return int(s,2)
%o A374625 print([a(n) for n in range(0,51)])
%o A374625 (Python)
%o A374625 def a(n):
%o A374625   d = {'0': '10', '1': '01'}
%o A374625   return int(''.join(map(d.get, bin(n)[2:])), 2)
%o A374625 print([a(n) for n in range(0,51)]) # _Jason Yuen_, Jul 15 2024
%o A374625 (Python)
%o A374625 a = lambda n: int(''.join("101"[b=='1':(b=='1')+2] for b in bin(n)[2:]), 2)
%o A374625 # _Peter Luschny_, Jul 15 2024
%o A374625 (Python)
%o A374625 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
%Y A374625 Cf. A030101, A115637, A179888.
%K A374625 nonn,base,easy
%O A374625 0,1
%A A374625 _Darío Clavijo_, Jul 14 2024