A104326 -id:A104326 - cp's OEIS frontend

A350312 Narayana weighted representation of n (the bottom version). Also binary representation of numbers not containing 00 or 010 as a substring.

0, 1, 10, 11, 101, 110, 111, 1011, 1101, 1110, 1111, 10110, 10111, 11011, 11101, 11110, 11111, 101101, 101110, 101111, 110110, 110111, 111011, 111101, 111110, 111111, 1011011, 1011101, 1011110, 1011111, 1101101, 1101110, 1101111, 1110110, 1110111, 1111011

Offset: 0

A.H.M. Smeets, Dec 24 2021

a(n) equals binary representation of m, if and only if A350311(m) = n and for all k < m, A350311(k) < n.

A.H.M. Smeets, The design of greedy number representations

Cf. A000930, A048715, A350215 (top version), A350311.

Fibonacci representations: A014417 (Zeckendorf), A104326 (dual Zeckendorf).

q[n_] := SequenceCount[IntegerDigits[n, 2], #] & /@ {{0, 0}, {0, 1, 0}} == {0, 0}; bin[n_] := FromDigits[IntegerDigits[n, 2]]; bin /@ Select[Range[0, 120], q] (* Amiram Eldar, Jan 27 2022 *)

# first method (as from definition)
def A101(n):
    f0, f1, f2, r = 1, 1, 1, 0
    while n > 0:
        if n%2 == 1:
            r = r+f0
        n, f0, f1, f2 = n//2, f0+f2, f0, f1
    return r
n, a = 0, 0
while n < 36:
    if A101(a) == n:
        print(bin(a)[2:], end = ", ")
        n += 1
    a += 1

# second method (as from regular expression)
def test(n):
    s, i, n1 = bin(n)[2:], 0, 2
    while i < len(s):
        if s[i] == "0":
            if n1 < 2:
                return 0
            n1 = 0
        else:
            n1 += 1
        i += 1
    return 1
n, a = 0, 0
while n < 36:
    if test(a):
        print(bin(a)[2:], end = ", ")
        n += 1
    a += 1

Regular expression: 0|11*(0111*)*(0|01|011*)?.

A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 15, 17, 13, 16, 14, 18, 19, 20, 25, 22, 28, 30, 21, 26, 29, 23, 27, 24, 31, 32, 33, 41, 46, 36, 43, 38, 49, 51, 34, 42, 37, 47, 50, 35, 44, 48, 39, 45, 40, 52, 53, 54, 67, 59, 75, 80, 56, 70, 77, 62, 72, 64, 83, 85, 55

Offset: 0

Rémy Sigrist, Aug 26 2022

This sequence is a self-inverse permutation of the nonnegative integers, similar to A345201 and A356331.

The dual Zeckendorf (or lazy Fibonacci) representation expresses uniquely a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary, and the corresponding binary encoding, A003754(n+1), cannot have two consecutive nonleading 0's.

			For n = 49:
- the dual Zeckendorf representation of 49 is "1111010",
- reversing its odd part ("111101"), we obtain "1011110",
- so a(49) = 39.

Rémy Sigrist, Table of n, a(n) for n = 0..10945
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A003754, A022290, A057889, A104326, A345201, A356331.

PARI
```
See Links section.
```

a(a(n)) = n.

a(n) < A000045(k) iff n < A000045(k).

A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 5, 4, 3, 7, 9, 8, 6, 10, 12, 16, 14, 11, 13, 17, 21, 27, 24, 19, 15, 22, 29, 35, 45, 40, 32, 18, 25, 37, 48, 58, 74, 66, 53, 20, 30, 42, 61, 79, 95, 121, 108, 87, 23, 33, 50, 69, 100, 129, 155, 197, 176, 142, 26, 38, 55, 82, 113, 163, 210, 252, 320, 286, 231

Offset: 1

A.H.M. Smeets, May 03 2024

The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326).
Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition.
Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1.
As a sequence, the array is a permutation of the nonnegative integers.
As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind.
In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers.

			Array begins:
       k=1    2    3    4    5    6    7
      +---------------------------------
  n=1 |  0    1    3    6   11   19   32
  n=2 |  2    4    8   14   24   40   66
  n=3 |  5    9   16   27   45   74  121
  n=4 |  7   12   21   35   58   95  155
  n=5 | 10   17   29   48   79  129  210
  n=6 | 13   22   37   61  100  163  265
  n=7 | 15   25   42   69  113  184  299
The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011:
          k=1      2       3        4
      +------------------------------
  n=1 |     0      1      11      111
  n=2 |    10    101    1011    10111
  n=3 |   110   1101   11011   110111
  n=4 |  1010  10101  101011  1010111
  n=5 |  1110  11101  111011  1110111

Cf. A104326.
Rows n=1..3: A001911, A019274, A014739.
Columns k=1..3: A090909, A276885, A188012.
Cf. k-th prepended column: A022342 (k=1), A023444 (k=2).

T(n,1) = A090909(n+1).
T(1,k) = A001911(k-1).
T(2,k) = A019274(k-2).
T(3,k) = A014739(k-1).
T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array.

A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

Original entry on oeis.org

3, 4, 5, 7, 15, 22, 25, 26, 27, 35, 41, 47, 49, 50, 73, 74, 75, 87, 89, 95, 97, 98, 101, 107, 121, 122, 135, 145, 146, 147, 167, 193, 194, 195, 207, 215, 217, 218, 221, 227, 241, 242, 255, 275, 289, 290, 315, 327, 335, 337, 338, 347, 361, 362, 385, 386, 387, 395

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

Numbers of the form F(k) - 1 have the same Zeckendorf (A014417) and dual Zeckendorf (A104326) representations: alternating digits of 1 and 0 whose sum is floor((k - 1)/2). Thus, if k is in this sequence then F(k) - 1 is both a Zeckendorf-Niven number (A328208) and a lazy-Fibonacci-Niven number (A328212), i.e., A000071(a(n)) is in A330711.

			7 is in this sequence since F(7) - 1 = 13 - 1 = 12 is divisible by floor((7 - 1)/2) = 3. The Zeckendorf and dual Zeckendorf representations of 7 are both 1010, whose sum of digits, 2, divides 12. Thus 12 is both a Zeckendorf-Niven number and a lazy-Fibonacci-Niven number.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A000071, A328208, A328212, A330711.

Select[Range[3, 400], Divisible[Fibonacci[#] - 1, Floor[(# - 1)/2]] &]

A328940 Numbers k such that k divides A003754(k+1).

Original entry on oeis.org

1, 2, 3, 23, 31, 61, 62, 173075, 259698, 332429, 2147535, 21217059, 72517101

Offset: 1

Amiram Eldar, Oct 31 2019

Numbers that divide the value of their dual Zeckendorf representation (A104326) when read as a binary number.
Analogous to A276488, with dual Zeckendorf representation instead of Zeckendorf representation (A014417).
The corresponding values of A003754(k+1) are 1, 2, 3, 46, 62, 183, 186, 15576750, 28826478, 45542773, 534736215, 15934011309, 100218633582, ... and the corresponding quotients are 1, 1, 1, 2, 2, 3, 3, 90, 111, 137, 249, 751, 1382, ...
a(14) > 3*10^9, if it exists.

			23 is in the sequence since the dual Zeckendorf representation of 23 is 101110 that equals 46 when read as a binary number, and 23|46.

Cf. A003754, A014417, A104326, A276488.

fb[n_] := Module[{k = Ceiling[Log[GoldenRatio, n * Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; fr];
dz[n_] := Module[{v = fb[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i+1]] == 0 && v[[i+2]] == 0, v[[i]] = 0; v[[i+1]] = 1; v[[i+2]] = 1; If[i>2, i-=3]]; i++];i=Position[v,_?(#>0&)]; If[i=={},{0}, v[[i[[1,1]];;-1]]]];
aQ[n_] := Divisible[FromDigits[dz[n], 2], n]; Select[Range[100], aQ]
(* after Robert G. Wilson v at A014417 and Ron Knott's Maple code at A104326 *)

cp's OEIS Frontend

A350312 Narayana weighted representation of n (the bottom version). Also binary representation of numbers not containing 00 or 010 as a substring.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A328940 Numbers k such that k divides A003754(k+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica