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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 1, 7, 0, 1, 2, 3, 12, 0, 1, 2, 0, 4, 5, 6, 20, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 33, 0, 1, 2, 0, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 13, 17, 18, 19, 54, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 12, 0, 1, 2, 0, 4, 5, 6, 20, 21, 22, 23, 24
Offset: 0

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Author

Rémy Sigrist, Aug 27 2022

Keywords

Comments

The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):
- in the Zeckendorf representation (or greedy Fibonacci representation):
- Fibonacci numbers are as big as possible (see A035517),
- and the corresponding binary encoding, A003714(n),
cannot have two consecutive 1's;
- in the dual Zeckendorf representation (or lazy Fibonacci representation):
- Fibonacci numbers are as small as possible (see A112309),
- and the corresponding binary encoding, A003754(n+1),
cannot have two consecutive nonleading 0's.
See A356326 for a similar sequence.

Examples

			For n = 28:
- using F(k) = A000045(k),
- the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
- the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
- F(5) and F(3) appear in both representations,
- so a(28) = F(5) + F(3) = 7.

Links

Crossrefs

Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.

Programs

  • PARI
    \\ See Links section.

Formula

a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).
a(n) = 0 iff n belongs to A331467.
a(n) = n iff n belongs to A000071.

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

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Aug 03 2022

Keywords

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

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.

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

Views

Author

Rémy Sigrist, Aug 03 2022

Keywords

Comments

This sequence has similarities with A334348.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    See Links section.

Formula

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).
Showing 1-3 of 3 results.

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