A309076 -id:A309076 - cp's OEIS frontend

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

Rémy Sigrist, Aug 03 2022

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

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

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

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

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

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

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

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.

A356325 Array A(n, k), n, k >= 0, read by antidiagonals; the terms in the negaFibonacci representation of A(n, k) are the terms in common in the negaFibonacci representations of n and k.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Aug 03 2022

This sequence has similarities with A334348.

			Array A(n, k) begins:
  n\k|  0  1  2  3   4  5  6  7  8   9  10  11  12  13
  ---+------------------------------------------------
    0|  0  0  0  0   0  0  0  0  0   0   0   0   0   0
    1|  0  1  0  1   0  0  1  0  1   0   0   1   0   0
    2|  0  0  2  2   0  0  0  2  2   0   0   0   0   0
    3|  0  1  2  3   0  0  1  2  3   0   0   1   0   0
    4|  0  0  0  0   4  5  5  5  5  -1   0   0  -1   0
    5|  0  0  0  0   5  5  5  5  5   0   0   0   0   0
    6|  0  1  0  1   5  5  6  5  6   0   0   1   0   0
    7|  0  0  2  2   5  5  5  7  7   0   0   0   0   0
    8|  0  1  2  3   5  5  6  7  8   0   0   1   0   0
    9|  0  0  0  0  -1  0  0  0  0   9  10  10  12  13
   10|  0  0  0  0   0  0  0  0  0  10  10  10  13  13
   11|  0  1  0  1   0  0  1  0  1  10  10  11  13  13
   12|  0  0  0  0  -1  0  0  0  0  12  13  13  12  13
   13|  0  0  0  0   0  0  0  0  0  13  13  13  13  13
.
For n = 14 and k = 43:
- using F(-k) = A039834(k):
- 14 = F(-1) + F(-7),
- 43 = F(-2) + F(-4) + F(-7) + F(-9),
- so A(14, 43) = F(-7) = 13.

Rémy Sigrist, Colored representation of the array for n, k <= 1000 (white for 0's, shades of blue for negative values, shades of red for positive values)
Rémy Sigrist, PARI program
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A004198, A039834, A215024, A309076, A334348, A356326, A356327.

PARI
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See Links section.
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A(n, k) = A(k, n).
A(n, n) = n.
A(n, 0) = 0.
A(n, k) = A356327(A215024(n) AND A215024(k)) (where AND denotes the bitwise AND operator).

cp's OEIS Frontend

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

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