A022290 -id:A022290 - cp's OEIS frontend

A374394 Irregular table T(n, k), n >= 0, 0 <= k < A277561(1+A003754(n)), read by rows; the n-th row lists the numbers z <= n such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

0, 0, 1, 0, 2, 1, 2, 0, 1, 3, 4, 2, 3, 2, 4, 0, 2, 5, 7, 1, 2, 6, 7, 3, 4, 5, 6, 3, 7, 4, 7, 0, 1, 3, 4, 8, 9, 11, 12, 2, 3, 10, 11, 2, 4, 10, 12, 5, 7, 8, 10, 6, 7, 9, 10, 5, 6, 11, 12, 7, 11, 7, 12, 0, 2, 5, 7, 13, 15, 18, 20, 1, 2, 6, 7, 14, 15, 19, 20, 3, 4, 5, 6, 16, 17, 18, 19

Offset: 0

Rémy Sigrist, Jul 07 2024

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------
   0  0
   1  0, 1
   2  0, 2
   3  1, 2
   4  0, 1, 3, 4
   5  2, 3
   6  2, 4
   7  0, 2, 5, 7
   8  1, 2, 6, 7
   9  3, 4, 5, 6
  10  3, 7
  11  4, 7
  12  0, 1, 3, 4, 8, 9, 11, 12
  13  2, 3, 10, 11
  14  2, 4, 10, 12

Rémy Sigrist, Table of n, a(n) for n = 0..8190
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A277561, A374354, A374395, A374396.

PARI
```
\\ See Links section.
```

T(n, k) = A022290(A374354(1+A003754(n)), k).

A374395 a(n) is the first term in the n-th row of A374394.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 0, 1, 3, 3, 4, 0, 2, 2, 5, 6, 5, 7, 7, 0, 1, 3, 3, 4, 8, 10, 10, 8, 9, 11, 11, 12, 0, 2, 2, 5, 6, 5, 7, 7, 13, 14, 16, 16, 17, 13, 15, 15, 18, 19, 18, 20, 20, 0, 1, 3, 3, 4, 8, 10, 10, 8, 9, 11, 11, 12, 21, 23, 23, 26, 27, 26, 28, 28, 21

Offset: 0

Rémy Sigrist, Jul 10 2024

a(n) is the least number z >= 0 such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A374355, A374394, A374396.

PARI
```
\\ See Links section.
```

a(n) = A022290(A374355(1+A003754(n)), k).

a(n) = n - A374396(n).

A374396 a(n) is the last term in the n-th row of A374394.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 4, 7, 7, 6, 7, 7, 12, 11, 12, 10, 10, 12, 11, 12, 20, 20, 19, 20, 20, 17, 16, 17, 20, 20, 19, 20, 20, 33, 32, 33, 31, 31, 33, 32, 33, 28, 28, 27, 28, 28, 33, 32, 33, 31, 31, 33, 32, 33, 54, 54, 53, 54, 54, 51, 50, 51, 54, 54, 53, 54, 54, 46

Offset: 0

Rémy Sigrist, Jul 10 2024

a(n) is the greatest number z <= n such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A374356, A374394, A374395.

PARI
```
\\ See Links section.
```

a(n) = A022290(A374356(1+A003754(n)), k).

a(n) = n - A374395(n).

A382113 Gray code transformation of the Zeckendorf representation of n.

Original entry on oeis.org

0, 1, 3, 6, 5, 11, 10, 8, 19, 18, 16, 13, 14, 32, 31, 29, 26, 27, 21, 22, 24, 53, 52, 50, 47, 48, 42, 43, 45, 34, 35, 37, 40, 39, 87, 86, 84, 81, 82, 76, 77, 79, 68, 69, 71, 74, 73, 55, 56, 58, 61, 60, 66, 65, 63, 142, 141, 139, 136, 137, 131, 132, 134, 123, 124

Offset: 0

Jeffrey Shallit, Mar 16 2025

			For n = 5: its Zeckendorf representation is 1000; the Gray code equivalent is 1111, which evaluates to 1+2+3+5=11. So a(5) = 11.

Cf. A003714, A006068, A022290, A382112.

Cf. A382116 (terms sorted).

a(n) = A022290(A006068(A003714(n))). In other words, take n, calculate its Zeckendorf representation, find the Gray code equivalent of that binary string; then regard it as the Zeckendorf representation of a number (even though it might have two consecutive 1's).

A356875 Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.

Original entry on oeis.org

1, 2, 5, 4, 10, 9, 8, 20, 18, 17, 16, 40, 36, 34, 21, 32, 80, 72, 68, 42, 33, 64, 160, 144, 136, 84, 66, 37, 128, 320, 288, 272, 168, 132, 74, 41, 256, 640, 576, 544, 336, 264, 148, 82, 65, 512, 1280, 1152, 1088, 672, 528, 296, 164, 130, 69, 1024, 2560, 2304, 2176, 1344, 1056, 592, 328, 260, 138, 73

Offset: 0

Peter Munn, Sep 02 2022

The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.
Write the first term in each row as Sum_{i in S} 2^i, where S is a set of nonnegative integers, then n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i).
More generally, if the terms are represented in binary, and the binary weighting of the digits (2^0, 2^1, 2^2, ...) is replaced with Fibonacci weighting (F_0, F_1, F_2, ...), we get the extended Wythoff array (A287870). If the weighting of the Zeckendorf representation is used (F_2, F_3, F_4, ...), we get the (unextended) Wythoff array (A035513).

			Square array A(n,k) begins:
   1    2    4    8    16    32    64   128 ...
   5   10   20   40    80   160   320   640 ...
   9   18   36   72   144   288   576  1152 ...
  17   34   68  136   272   544  1088  2176 ...
  21   42   84  168   336   672  1344  2688 ...
  33   66  132  264   528  1056  2112  4224 ...
  37   74  148  296   592  1184  2368  4736 ...
  41   82  164  328   656  1312  2624  5248 ...
  65  130  260  520  1040  2080  4160  8320 ...
  69  138  276  552  1104  2208  4416  8832 ...
  ...
The defining characteristic of a Fibbinary number is that its binary representation does not have a 1 followed by another 1. Shown in binary the array begins:
      1      10      100      1000 ...
    101    1010    10100    101000 ...
   1001   10010   100100   1001000 ...
  10001  100010  1000100  10001000 ...
  10101  101010  1010100  10101000 ...
  ...

See the comments for the relationship to: A000045, A003714, A035513, A054582, A287870.

See the formula section for the relationship to: A022290, A022341, A356874.

A(n,0) = A022341(n), otherwise A(n,k) = 2*A(n,k-1).
A287870(n+1,k+1) = A356874(floor(A(n,k)/2)).
A035513(n+1,k+1) = A022290(A(n,k)).

A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Sep 06 2022

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation.

See A334348 for the sequence dealing with Zeckendorf (or greedy Fibonacci) representations. Unlike A334348, the present sequence is not associative.

			Square 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  1  0  1  0  1  1   0   1   1   0
    2|  0  0  2  2  0  2  2  2  2  0   2   2   0   2
    3|  0  1  2  3  1  2  3  2  3  1   2   3   1   2
    4|  0  1  0  1  4  3  4  0  1  4   3   4   4   3
    5|  0  0  2  2  3  5  5  2  2  3   5   5   3   5
    6|  0  1  2  3  4  5  6  2  3  4   5   6   4   5
    7|  0  0  2  2  0  2  2  7  7  5   7   7   0   2
    8|  0  1  2  3  1  2  3  7  8  6   7   8   1   2
    9|  0  1  0  1  4  3  4  5  6  9   8   9   4   3
   10|  0  0  2  2  3  5  5  7  7  8  10  10   3   5
   11|  0  1  2  3  4  5  6  7  8  9  10  11   4   5
   12|  0  1  0  1  4  3  4  0  1  4   3   4  12  11
   13|  0  0  2  2  3  5  5  2  2  3   5   5  11  13

Rémy Sigrist, Colored representation of the table for x, y < Fibonacci(16)-1 (the hue is function of A(x,y))
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A334348.

PARI
```
See Links section.
```

A(n, k) = A022290(A003754(n+1) AND A003754(k+1)) (where AND denotes the bitwise AND operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, n) = n.

A197432 a(n) = Sum_{k>=0} A030308(n,k)*C(k) where C(k) is the k-th Catalan number (A000108).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 42, 43, 43, 44, 44, 45, 45, 46, 47, 48, 48, 49, 49, 50, 50, 51, 56, 57, 57, 58, 58, 59, 59, 60, 61, 62, 62, 63, 63, 64, 64, 65

Offset: 0

Philippe Deléham, Oct 15 2011

Replace 2^k with A000108(k) in binary expansion of n.

			11 = 1011_2, so a(11) = 1*1 + 1*1 + 0*2 + 1*5 = 7.

Cf. A000108, A030308, A197433.

Other sequences that are built by replacing 2^k in binary representation with other numbers: A022290 (Fibonacci), A029931 (natural numbers), A059590 (factorials), A089625 (primes), A197354 (odd numbers).

G.f.: (1/(1 - x))*Sum_{k>=0} Catalan number(k)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017

A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 8, 16, 24, 48, 40, 80, 120, 240, 13, 26, 39, 78, 65, 130, 195, 390, 104, 208, 312, 624, 520, 1040, 1560, 3120, 21, 42, 63, 126, 105, 210, 315, 630, 168, 336, 504, 1008, 840, 1680, 2520, 5040, 273, 546, 819, 1638, 1365, 2730, 4095, 8190

Offset: 0

Ilya Gutkovskiy, Aug 19 2019

nonn

			23 = 2^0 + 2^1 + 2^2 + 2^4 so a(23) = Fibonacci(3) * Fibonacci(4) * Fibonacci(5) * Fibonacci(7) = 390.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta additionally represents powerful numbers that are not perfect prime powers and bright green represents primorials.

Cf. A000045, A003266, A019565, A022290, A029930, A121663, A160009.

nmax = 55; CoefficientList[Series[Product[(1 + Fibonacci[k + 3] x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Fibonacci[Floor[Log[2, n]] + 3] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]

a(n)={vecprod([fibonacci(k+2) | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

G.f.: Product_{k>=0} (1 + Fibonacci(k + 3) * x^(2^k)).
a(0) = 1; a(n) = Fibonacci(floor(log_2(n)) + 3) * a(n - 2^floor(log_2(n))).
a(2^(k-2)-1) = A003266(k).

A326032 a(2^x + ... + 2^z) = w(x) + ... + w(z), where x...z are distinct nonnegative integers and w = A000120.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 3, 3, 4, 4, 4, 4, 5

Offset: 0

Gus Wiseman, Jul 22 2019

nonn

From Robert Israel, Jul 23 2019: (Start)
a(2*n+1)=a(2*n).
a(n)=1 if and only if n > 1 is in A283526. (End)

			For example, a(6) = a(2^2 + 2^1) = w(2) + w(1) = 2.

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A022290 (Fibonacci), A059590 (factorials), A073642, A089625 (primes), A116549, A326031.

Cf. A000120, A029931, A035327, A048793, A070939, A283526, A305830, A326031, A326669, A326702.

Bwt:= proc(n) option remember; convert(convert(n,base,2),`+`) end proc:
f:= proc(n) local L,i;
  L:= convert(n,base,2);
  add(L[i]*Bwt(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Jul 23 2019

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[Length/@bpe/@(bpe[n]-1)],{n,0,100}]

A361789 A(n, k) is the sum of the distinct terms in the dual Zeckendorf representations of n or of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 6, 6, 3, 6, 6, 6, 7, 6, 5, 6, 6, 5, 6, 7, 8, 8, 6, 6, 4, 6, 6, 8, 8, 9, 8, 7, 6, 6, 6, 6, 7, 8, 9, 10, 9, 8, 8, 6, 5, 6, 8, 8, 9, 10, 11, 11, 11, 8, 11, 6, 6, 11, 8, 11, 11, 11, 12, 11, 10, 11, 11, 10, 6, 10, 11, 11, 10, 11, 12

Offset: 0

Rémy Sigrist, Mar 24 2023

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation (see A356771 for further details).

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
  ----+-------------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
    1 |  1   1   3   3   4   6   6   8   8   9  11  11  12  14
    2 |  2   3   2   3   6   5   6   7   8  11  10  11  14  13
    3 |  3   3   3   3   6   6   6   8   8  11  11  11  14  14
    4 |  4   4   6   6   4   6   6  11  11   9  11  11  12  14
    5 |  5   6   5   6   6   5   6  10  11  11  10  11  14  13
    6 |  6   6   6   6   6   6   6  11  11  11  11  11  14  14
    7 |  7   8   7   8  11  10  11   7   8  11  10  11  19  18
    8 |  8   8   8   8  11  11  11   8   8  11  11  11  19  19
    9 |  9   9  11  11   9  11  11  11  11   9  11  11  17  19
   10 | 10  11  10  11  11  10  11  10  11  11  10  11  19  18
   11 | 11  11  11  11  11  11  11  11  11  11  11  11  19  19
   12 | 12  12  14  14  12  14  14  19  19  17  19  19  12  14
   13 | 13  14  13  14  14  13  14  18  19  19  18  19  14  13

Rémy Sigrist, Colored representation of the table for x, y < Fibonacci(16) - 1 (where the color is function of A(x, y))
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A003986, A022290, A334348, A356771, A356969, A361756.

PARI
```
See Links section.
```

A(n, k) = A022290(A003754(n+1) OR A003754(k+1)) (where OR denotes the bitwise OR operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, n) = n.
A(A(m, n), k) = A(m, A(n, k)).
A(A(n, k), n) = A(n, k).
A(n, A361756(n, k)) = n.

cp's OEIS Frontend

A374394 Irregular table T(n, k), n >= 0, 0 <= k < A277561(1+A003754(n)), read by rows; the n-th row lists the numbers z <= n such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A374395 a(n) is the first term in the n-th row of A374394.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A374396 a(n) is the last term in the n-th row of A374394.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A382113 Gray code transformation of the Zeckendorf representation of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A356875 Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A197432 a(n) = Sum_{k>=0} A030308(n,k)*C(k) where C(k) is the k-th Catalan number (A000108).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A326032 a(2^x + ... + 2^z) = w(x) + ... + w(z), where x...z are distinct nonnegative integers and w = A000120.

Views

Author

Keywords