A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

1, 2, 2, 2, 4, 4, 2, 2, 4, 8, 6, 4, 6, 4, 2, 2, 4, 8, 10, 8, 12, 12, 6, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 12, 24, 20, 12, 16, 12, 6, 4, 6, 12, 16, 12, 16, 16, 8, 4, 6, 12, 8, 4, 6, 4, 2, 2, 4, 8, 10, 8, 16, 20, 10, 8, 16, 32, 28, 20, 28

Offset: 0

Author

Rémy Sigrist, Jun 17 2023

Keywords

• nonn
• base

Comments

Equivalently, a(n) is the number of k >= 0 such that A332497(k) + A332498(k) = n.

The set of pairs of nonnegative integers (x, y) such that A003188(x) AND A003188(y) = 0 is related to the T-square fractal (see illustration in Links section).

Examples

			For n = 8:
- we have:
  k  A332497(8-k)  A332497(k)  A332497(8-k) AND A332497(k)
  -  ------------  ----------  ---------------------------
  0            12           0                            0
  1             4           1                            0
  2             5           3                            1
  3             7           2                            2
  4             6           6                            6
  5             2           7                            2
  6             3           5                            1
  7             1           4                            0
  8             0          12                            0
- so a(8) = 4.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Scatterplot of (x, y) such that x, y < 2^10 and A003188(x) AND A003188(y) = 0
• Wikipedia, T-square (fractal)

Crossrefs

Cf. A003188, A075427, A332497, A332498.

Programs

PARI

a(n) = 2*sum(k=0, n\2, bitand(bitxor(n-k, (n-k)\2), bitxor(k, k\2))==0) - (n==0)

Python

A363710=lambda n: sum(map(lambda k: not (k^k>>1)&(n-k^n-k>>1),range(n+1>>1)))<<1 if n else 1 # Natalia L. Skirrow, Jun 22 2023

Formula

a(n) = 2 iff n belongs to A075427.