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

0,2

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).

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)

Formula

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

Example

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.

Mathematica

A131218[n_, k_]:= Boole[BitAnd[BitXor[n, BitShiftRight[n, 1]], BitXor[k, BitShiftRight[k, 1]]] ==0];
A363710[n_]:= A363710[n]= Sum[A131218[n-k, k], {k, 0, n}];
Table[A363710[n], {n, 0, 100}] (* G. C. Greubel, Sep 06 2025 *)

Prog

(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
(Magma)
A131218:= func< n, k | BitwiseAnd(BitwiseXor(n, ShiftRight(n, 1)), BitwiseXor(k, ShiftRight(k, 1))) eq 0 select 1 else 0 >;
A363710:= func< n | (&+[A131218(n-k, k): k in [0..n]]) >;
[A363710(n): n in [0..100]]; // G. C. Greubel, Sep 06 2025
(SageMath)
def A363710(n): return sum(int( ((n-k)^^((n-k)>>1)) & (k^^(k>>1)) ==0) for k in range(n+1))
print([A363710(n) for n in range(101)]) # G. C. Greubel, Sep 06 2025

Crossrefs

Cf. A003188, A075427, A131218, A332497, A332498.

Sequence in context: A389763 A139560 A192095 * A380272 A105080 A336299

Adjacent sequences: A363707 A363708 A363709 * A363711 A363712 A363713

Keyword

nonn,base

Author

Rémy Sigrist, Jun 17 2023

Status

approved