A356327 Replace 2^k in binary expansion of n with A039834(1+k).

0, 1, -1, 0, 2, 3, 1, 2, -3, -2, -4, -3, -1, 0, -2, -1, 5, 6, 4, 5, 7, 8, 6, 7, 2, 3, 1, 2, 4, 5, 3, 4, -8, -7, -9, -8, -6, -5, -7, -6, -11, -10, -12, -11, -9, -8, -10, -9, -3, -2, -4, -3, -1, 0, -2, -1, -6, -5, -7, -6, -4, -3, -5, -4, 13, 14, 12, 13, 15, 16

Offset: 0

Author

Rémy Sigrist, Aug 03 2022

Keywords

• sign
• base

Comments

This sequence has similarities with A022290, and is related to negaFibonacci representations.

Examples

			For n = 13:
- 13 = 2^3 + 2^2 + 2^0,
- so a(13) = A039834(4) + A039834(3) + A039834(1) = -3 + 2 + 1 = 0.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Wikipedia, NegaFibonacci coding
• Index entries for sequences related to Zeckendorf expansion of n

Crossrefs

Cf. A000201, A001950, A003714, A004957, A020989, A022290, A026351, A039834, A060144, A072197, A189663, A215024, A215025, A309076, A356325, A356326.

Programs

Mathematica

Table[Reverse[#].Fibonacci[-Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 69}] (* Rémy Sigrist, Aug 05 2022 *)

PARI

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=fibonacci(-1-k)); return (v) }

Python

from sympy import fibonacci
def A356327(n): return sum(fibonacci(-a)*int(b) for a, b in enumerate(bin(n)[:1:-1],start=1)) # Chai Wah Wu, Aug 31 2022

Formula

a(n) = Sum_{k>=0} A030308(n,k)*A039834(1+k).

a(A215024(n)) = n.

a(A215025(n)) = -n.

a(A003714(n)) = A309076(n).

Empirically:

- a(n) = 0 iff n = 0 or n belongs to A072197,

- a(n) = 1 iff n belongs to A020989,

- a(2*A215024(n)) = -A000201(n) for n > 0,

- a(3*A215024(n)) = -A060143(n),

- a(floor(A215024(n)/2)) = -A060143(n),

- a(4*A215024(n)) = A001950(n) for n > 0,

- a(floor(A215024(n)/4)) = A189663(n) for n > 0,

- a(2*A215025(n)) = A026351(n),

- a(3*A215025(n)) = A019446(n) for n > 0,

- a(floor(A215025(n)/2)) = A019446(n) for n > 0,

- a(4*A215025(n)) = -A004957(n),

- a(floor(A215025(n)/4)) = -A060144(n+1) for n >= 0.