A331216 a(n) is the number of ways to write n = u + v where the binary representations of u and of v have the same number of 0's and the same number of 1's.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 2, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 4, 3, 2, 5, 2, 3, 6, 3, 2, 5, 2, 3, 4, 3, 0, 3, 2, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 3, 4, 3, 2, 5, 2, 3, 6, 3, 4, 7, 2, 7, 6, 5

Offset: 0

Author

Rémy Sigrist, Jan 12 2020

Keywords

• nonn
• base

Comments

In other words, a(n) is the number of ways to write n as the sum of two binary anagrams.

Leading zeros are ignored.

Examples

			For n = 22:
- we can write 22 as u + v in the following ways:
  u   v   bin(u)  bin(v)
  --  --  ------  ------
  10  12    1010    1100
  11  11    1011    1011
  12  10    1100    1010
- hence a(22) = 3.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16384
• Rémy Sigrist, PARI program for A331216
• Rémy Sigrist, Scatterplot of (x, y) such that 0 <= x, y <= 2^10 and x and y are binary anagrams (a(n) corresponds to the number of pixels (x, y) such that x+y = n)

Crossrefs

Cf. A330827 (ternary analog), A331218 (decimal analog).

Programs

PARI

See Links section.

Formula

a(2*n) > 0.

a(2*n) >= a(n).

Apparently, a(3*2^k-1-x) = a(3*2^k-1+x) for any k >= 0 and x = -2^k..2^k.