A361398 An infiltration of two words, say x and y, is a shuffle of x and y optionally followed by replacements of pairs of consecutive equal symbols, say two d's, one of which comes from x and the other from y, by a single d (that cannot be part of another replacement); a(n) is the number of distinct infiltrations of the word given by the binary representation of n with itself.

1, 2, 5, 3, 9, 12, 9, 4, 14, 28, 30, 21, 19, 21, 14, 5, 20, 53, 68, 60, 55, 74, 68, 32, 34, 60, 55, 36, 34, 32, 20, 6, 27, 89, 126, 134, 120, 181, 196, 108, 88, 181, 183, 136, 151, 164, 126, 45, 55, 134, 151, 129, 107, 136, 120, 54, 69, 108, 88, 54, 55, 45, 27

Offset: 0

Author

Rémy Sigrist, Mar 10 2023

Keywords

• nonn
• base

Comments

Leading zeros in binary expansions are ignored.

See A191755 for the definition of a shuffle.

Examples

			For n = 2:
- the binary expansion of 2 is "10",
- we have essentially the following infiltrations:
         x        10   10    1 0   10     1 0
         y        10   1 0    10     10    1 0
                  --   ---   ---   ----   ----
    infiltration  10   100   110   1010   1100
- so a(2) = 5.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, PARI program
• Wikipedia, Infiltration product.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000096, A191755, A193020.

Programs

PARI

See Links section.

Formula

a(n) >= A193020(n).

a(2^k - 1) = k + 1 for any k >= 0.

a(2^k) = A000096(k + 1) for any k >= 0.