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.

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, 26, 10, 21, 1, 62, 30, 33, 14, 49, 17, 46, 6, 57, 25, 38, 9, 54, 22, 41, 2, 61, 29, 34, 13, 50, 18, 45, 5, 58, 26, 37, 10, 53, 21, 42, 1, 126, 62, 65, 30, 97, 33, 94

Offset: 0

Author

Darío Clavijo, Nov 22 2024

Keywords

• nonn
• base
• look

Comments

a(n) is Gray-coded into the reversed binary representation of n.

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).

Examples

			For n = 75 a(75) = 78 because:
75 in base 2 is 1001011 and in base 2:
  m      | x = x XOR (m AND 1) | o
---------+---------------------+----------
1001011  | 1 = 0 XOR 1         |       1
100101   | 0 = 1 XOR 1         |      10
10010    | 0 = 0 XOR 0         |     100
1001     | 1 = 0 XOR 1         |    1001
100      | 1 = 1 XOR 0         |   10011
10       | 1 = 1 XOR 0         |  100111
1        | 0 = 1 XOR 1         | 1001110
And 1001110 in base 10: 78
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..16383
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2^13-1, with a color function showing a(n) according to floor(log_2(a(n))).
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000079, A000225, A000051, A000975, A007283, A006068, A030101, A095121.

Cf. A380001 (fixed points).

Programs

Mathematica

A378299[n_] := FromDigits[FoldList[BitXor, 0, Reverse[IntegerDigits[n, 2]]], 2];
Array[A378299, 100, 0] (* Paolo Xausa, Dec 13 2024 *)

Python

def a(n):
  m,x,o = n,0,0
  while m > 0:
    x ^= (m & 1)
    o <<= 1
    o |= x
    m >>=1
  return o
print([a(n) for n in range(0,71)])

Formula

a(n) = A006068(A030101(n)).

a(A000079(n)) = 1.

a(A007283(n)) = 2.

a(A000225(n)) = A000975(n).

a(A000051(n)) = A095121(n).