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A377029 a(1) = 0; thereafter in the binary expansion of a(n-1), expand bits: 1->01 and 0->10.

%I A377029 #54 Jan 18 2025 21:58:11
%S A377029 0,2,6,22,406,92566,6818458006,26055178074437806486,
%T A377029 540213899028732737068658940860686756246,
%U A377029 163551003506862550406254063077517364557434408527734307437037618419534882498966
%N A377029 a(1) = 0; thereafter in the binary expansion of a(n-1), expand bits: 1->01 and 0->10.
%C A377029 All terms are even and leading zeros omitted in the final encoding.
%C A377029 Conversely the opposite mapping of bits: 0->01 and 1->10 is A133468.
%C A377029 The bit length of a(n) is 2^(n-1)+1.
%C A377029 The count of bits set for a(n) is A094373(n).
%C A377029 a(n) = 2 (mod 4) for n > 1.
%C A377029 Also all the terms align bitwise to the right.
%C A377029 The hamming distance of a(n) and a(n+1) is in A000079.
%H A377029 Darío Clavijo, <a href="/A377029/b377029.txt">Table of n, a(n) for n = 1..13</a>
%F A377029 a(n) = A320916(2^(n-2)+1) for n > 1.
%F A377029 A000120(a(n+1) XOR a(n)) = A000079(n-2).
%F A377029 a(n) = A374625(a(n-1)) for n > 1. - _Paolo Xausa_, Nov 04 2024
%e A377029 For n = 5 a(5) = 406 because:
%e A377029 This encoding results in the following tree:
%e A377029 n | a(n)
%e A377029 --+---------------
%e A377029 1 | 0
%e A377029   | |\
%e A377029 2 | 1 0
%e A377029   | | |
%e A377029 3 | 1 10
%e A377029   | | | \
%e A377029 4 | 1 01 10--
%e A377029   | | |\  \  \
%e A377029   | | | \  \  \
%e A377029 5 | 1 10 01 01 10
%e A377029 Which also aligns bitwise to the right:
%e A377029 n | a(n)
%e A377029 --+-----------
%e A377029 1 |         0
%e A377029 2 |        10
%e A377029 3 |       110
%e A377029 4 |     10110
%e A377029 5 | 110010110
%e A377029 And 110010110 in base 10 is 406.
%t A377029 NestList[FromDigits[2 - IntegerDigits[#, 2], 4] &, 0, 10] (* _Paolo Xausa_, Nov 04 2024 *)
%o A377029 (Python)
%o A377029 from functools import cache
%o A377029 A374625 = lambda n: int(bin(n)[2:].replace('0', '2'), 4)
%o A377029 @cache
%o A377029 def a(n):
%o A377029   if n == 1: return 0
%o A377029   return A374625(a(n-1))
%o A377029 print([a(n) for n in range(1, 12)])
%Y A377029 Cf. A000051, A000079, A000120, A094373, A133468, A320916, A374625.
%K A377029 nonn,base,easy
%O A377029 1,2
%A A377029 _Darío Clavijo_, Oct 13 2024