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A378299 Read the binary representation of n from the most to least significant bit then perform a cumulative XOR and store by reading from least to most significant bit.

%I A378299 #48 Jan 11 2025 14:06:02
%S A378299 0,1,1,2,1,6,2,5,1,14,6,9,2,13,5,10,1,30,14,17,6,25,9,22,2,29,13,18,5,
%T A378299 26,10,21,1,62,30,33,14,49,17,46,6,57,25,38,9,54,22,41,2,61,29,34,13,
%U A378299 50,18,45,5,58,26,37,10,53,21,42,1,126,62,65,30,97,33,94
%N A378299 Read the binary representation of n from the most to least significant bit then perform a cumulative XOR and store by reading from least to most significant bit.
%C A378299 a(n) is Gray-coded into the reversed binary representation of n.
%C A378299 Fixed points are 0 and f(n) = 8*f(n-1) + 5 with f(1)=1 or f(n) = (1/14)*(3*(2^(3*n))-10) for n >= 1 (cf. A380001).
%H A378299 Paolo Xausa, <a href="/A378299/b378299.txt">Table of n, a(n) for n = 0..16383</a>
%H A378299 Michael De Vlieger, <a href="/A378299/a378299.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A378299 Michael De Vlieger, <a href="/A378299/a378299_1.png">Fan style binary tree showing a(n)</a>, n = 0..2^13-1, with a color function showing a(n) according to floor(log_2(a(n))).
%H A378299 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A378299 a(n) = A006068(A030101(n)).
%F A378299 a(A000079(n)) = 1.
%F A378299 a(A007283(n)) = 2.
%F A378299 a(A000225(n)) = A000975(n).
%F A378299 a(A000051(n)) = A095121(n).
%e A378299 For n = 75 a(75) = 78 because:
%e A378299 75 in base 2 is 1001011 and in base 2:
%e A378299   m      | x = x XOR (m AND 1) | o
%e A378299 ---------+---------------------+----------
%e A378299 1001011  | 1 = 0 XOR 1         |       1
%e A378299 100101   | 0 = 1 XOR 1         |      10
%e A378299 10010    | 0 = 0 XOR 0         |     100
%e A378299 1001     | 1 = 0 XOR 1         |    1001
%e A378299 100      | 1 = 1 XOR 0         |   10011
%e A378299 10       | 1 = 1 XOR 0         |  100111
%e A378299 1        | 0 = 1 XOR 1         | 1001110
%e A378299 And 1001110 in base 10: 78
%t A378299 A378299[n_] := FromDigits[FoldList[BitXor, 0, Reverse[IntegerDigits[n, 2]]], 2];
%t A378299 Array[A378299, 100, 0] (* _Paolo Xausa_, Dec 13 2024 *)
%o A378299 (Python)
%o A378299 def a(n):
%o A378299   m,x,o = n,0,0
%o A378299   while m > 0:
%o A378299     x ^= (m & 1)
%o A378299     o <<= 1
%o A378299     o |= x
%o A378299     m >>=1
%o A378299   return o
%o A378299 print([a(n) for n in range(0,71)])
%Y A378299 Cf. A000079, A000225, A000051, A000975, A007283, A006068, A030101, A095121.
%Y A378299 Cf. A380001 (fixed points).
%K A378299 nonn,base,look
%O A378299 0,4
%A A378299 _Darío Clavijo_, Nov 22 2024