A342697 For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6, 7, 7, 7, 0, 0, 0, 1, 4, 5, 7, 7, 12, 12, 14, 15, 14, 15, 15, 15, 0, 0, 0, 1, 0, 1, 3, 3, 8, 8, 10, 11, 14, 15, 15, 15, 24, 24, 24, 25, 28, 29, 31, 31, 28, 28, 30, 31, 30, 31, 31, 31, 0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6

Offset: 0

Author

Rémy Sigrist, Mar 18 2021

Keywords

• nonn
• base
• easy
• look

Comments

The value of the k-th bit in a(n) corresponds to the most frequent value in the bit triple starting at the k-th bit in n.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     1     101          1
   6     3     110         11
   7     3     111         11
   8     0    1000          0
   9     0    1001          0
  10     2    1010         10
  11     3    1011         11
  12     6    1100        110
  13     7    1101        111
  14     7    1110        111
  15     7    1111        111
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^20.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A048715, A048730, A048733, A342698, A342700.

Programs

Mathematica

A342697[n_] := Quotient[7*n - BitXor[n, 2*n, 4*n], 8];
Array[A342697, 100, 0] (* Paolo Xausa, Aug 06 2025 *)

PARI

a(n) = sum(k=0, #binary(n), ((bittest(n, k)+bittest(n, k+1)+bittest(n, k+2))>=2) * 2^k)

Formula

a(n) = 0 iff n belongs to A048715.

a(n) = floor(A048730(n)/8) = floor(A048733(n)/2). - Kevin Ryde, Mar 26 2021