Peter G. Anderson - cp's OEIS frontend

A257962 List of successive permutations of the numbers [0,F(n)) achieved by writing the numbers in the Fibonacci/Zeckendorf radix in the form of an (n-2)-bit string, reversing that string, and extracting the numerical value according to Fibonacci/Zeckendorf radix.

0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 5, 3, 2, 7, 1, 6, 4, 0, 8, 5, 3, 11, 2, 10, 7, 1, 9, 6, 4, 12, 0, 13, 8, 5, 18, 3, 16, 11, 2, 15, 10, 7, 20, 1, 14, 9, 6, 19, 4, 17, 12, 0, 21, 13, 8, 29, 5, 26, 18, 3, 24, 16, 11, 32, 2, 23, 15, 10, 31, 7, 28, 20, 1, 22, 14

Offset: 0

Peter G. Anderson, May 14 2015

This sequence divides into blocks of length F(n), n = 1, 2, 3, 4, 5, ... (so F(n) = 1, 2, 3, 5, 8, ...)

			This is an irregular array, the first few rows of which are:
0;
0, 1;
0, 2, 1;
0, 3, 2, 1, 4;
0, 5, 3, 2, 7, 1, 6, 4;
0, 8, 5, 3, 11, 2, 10, 7, 1, 9, 6, 4, 12;
0, 13, 8, 5, 18, 3, 16, 11, 2, 15, 10, 7, 20, 1, 14, 9, 6, 19, 4, 17, 12;

Peter G. Anderson, Table of n, a(n) for n = 0..317808

Cf. A014417, A257691.

The function ztab in the j language computes a table of Zeckendorf representations for the integers [0,F(n+2))
ztab =: 3 : 0
        if. y = 1 do. 2 1 $ 0 1
        elseif. y = 2 do. 3 2 $ 0 0 0 1 1 0
        elseif. do. (0  ,"1 ztab y-1), (1 0 ,"1 ztab y-2)
        end.
)
   ztab 5
0 0 0 0 0
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 0 1 0 1
0 1 0 0 0
0 1 0 0 1
0 1 0 1 0
1 0 0 0 0
1 0 0 0 1
1 0 0 1 0
1 0 1 0 0
1 0 1 0 1
Then forming the inner product with the Fibonacci numbers reversed (i.e., evaluating in the Fibonacci radix backwards) give the permutation:
   1 2 3 5 8 +/ . * "1 ztab 5
0 8 5 3 11 2 10 7 1 9 6 4 12

A257961 List of permutations of the intervals of numbers [0,F(n)) defined by x -> x*F(n-1) mod F(n), where F(n) is the n-th Fibonacci number A000045.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 1, 4, 2, 0, 5, 2, 7, 4, 1, 6, 3, 0, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5, 0, 13, 5, 18, 10, 2, 15, 7, 20, 12, 4, 17, 9, 1, 14, 6, 19, 11, 3, 16, 8, 0, 21, 8, 29, 16, 3, 24, 11, 32, 19, 6, 27, 14, 1, 22, 9, 30, 17, 4, 25, 12, 33, 20, 7

Offset: 0

Peter G. Anderson, May 14 2015

This sequence divides into blocks of length F(n), n = 2, 3, 4, 5, 6, ... (so F(n) = 1, 2, 3, 5, 8, ...)

			This is an irregular array, the first few rows of which are:
0;
0, 1;
0, 2, 1;
0, 3, 1, 4, 2;
0, 5, 2, 7, 4, 1, 6, 3;
0, 8, 3, 11, 6, 1, 9, 4, 12, 7, 2, 10, 5;
0, 13, 5, 18, 10, 2, 15, 7, 20, 12, 4, 17, 9, 1, 14, 6, 19, 11, 3, 16, 8;

Peter G. Anderson, Table of n, a(n) for n = 0..317808

row(n) = if (n<=2, [0], vector(fibonacci(n), k, (k-1)*fibonacci(n-1) % fibonacci(n))); \\ Michel Marcus, May 28 2015

A215781 a(n) = ceiling(n*(sqrt(3)-1)).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 42

Offset: 0

Peter G. Anderson, Aug 23 2012

sqrt(3)-1 has continued fraction [0,1,2,1,2,1,2,...].

Peter G. Anderson, Table of n, a(n) for n = 0..499 (corrected by Ray Chandler, Jan 19 2019)

A188070 is ceiling(n/(sqrt(3)-1)).

J
```
>. ( _1 + %:3) * i.500
```

Table[Ceiling[n(Sqrt[3]-1)],{n,0,60}] (* Harvey P. Dale, Jan 15 2017 *)

Definition corrected by N. J. A. Sloane, Jan 15 2017

A215592 Value of y in the n-th number of the form x+y*(3+sqrt(13))/2.

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 16 2012

nonn

Differs from A025684 first at the index 308. (The 0 and 13 at indices 308 and 309 are swapped). - R. J. Mathar, Mar 10 2013

Peter G. Anderson, Table of n, a(n) for n = 1..584

A215591 is the value of x.

A215591 Value of x in the n-th number of the form x+y*(3+sqrt(13))/2.

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 16 2012

nonn

Differs from A025639 first at n=308 and n=309, next at n=322 and n=323. - R. J. Mathar, Aug 21 2012

Peter G. Anderson, Table of n, a(n) for n = 1..584

A215592 is the value of y.

A215590 Value of y in the n-th number of the form x+y*(1+sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 16 2012

nonn

A215589 gives the value of x.

Peter G. Anderson, Table of n, a(n) for n = 1..1064

Cf. A215589.

A215589 Value of x in the n-th number of the form x+y*(1+sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 16 2012

nonn

A215590 gives the value of y.

Peter G. Anderson, Table of n, a(n) for n = 1..1064

Cf. A215590.

A215345 Value of y in the n-th number of the form x+y*(1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 08 2012

nonn

Let x>=0, y>=0 be integers, sort according to x+y*(1+sqrt(5))/2, this sequence gives the y-values. - Joerg Arndt, Aug 16 2012

			Let g = (1+sqrt(5))/2, sequences A215344 (x) and A215345 (y) start as:
[x+y*g, x, y]
[0.0000000, 0, 0]
[1.0000000, 1, 0]
[1.6180340, 0, 1]
[2.0000000, 2, 0]
[2.6180340, 1, 1]
[3.0000000, 3, 0]
[3.2360680, 0, 2]
[3.6180340, 2, 1]
[4.0000000, 4, 0]
[4.2360680, 1, 2]
[4.6180340, 3, 1]
[4.8541020, 0, 3]
[5.0000000, 5, 0]
[5.2360680, 2, 2]
[5.6180340, 4, 1]
- _Joerg Arndt_, Aug 17 2012.

Peter G. Anderson, Table of n, a(n) for n = 1..1571

A215344 is the value of x.

Cf. A084531, A054065, A194832.

PARI
```
/* see A215344 */
```

A215344 Value of x in the n-th number of the form x+y*(1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Peter G. Anderson, Aug 08 2012

nonn

Let x>=0, y>=0 be integers, sort according to x+y*(1+sqrt(5))/2, this sequence gives the x-values. [Joerg Arndt, Aug 16 2012]

The positive elements of this sequence are A084531. [Peter G. Anderson, Aug 28 2012]

			Let g = (1+sqrt(5))/2, sequences A215344 (x) and A215345 (y) start as:
[x+y*g, x, y]
[0.0000000, 0, 0]
[1.0000000, 1, 0]
[1.6180340, 0, 1]
[2.0000000, 2, 0]
[2.6180340, 1, 1]
[3.0000000, 3, 0]
[3.2360680, 0, 2]
[3.6180340, 2, 1]
[4.0000000, 4, 0]
[4.2360680, 1, 2]
[4.6180340, 3, 1]
[4.8541020, 0, 3]
[5.0000000, 5, 0]
[5.2360680, 2, 2]
[5.6180340, 4, 1]
- _Joerg Arndt_, Aug 17 2012.

Peter G. Anderson, Table of n, a(n) for n = 1..1571

A215345 is the value of y.

default(realprecision,99); /* using floats */
g=(1+sqrt(5))/2; /* golden ratio */
M = 100.0;  /* search limit:  x + y * g <= M */
v=vector(ceil(M)^2);
ct=0;
{ for (x=0, M,
    for (y=0, (M-x)/g,
        ct += 1;
        v[ct] = [x+y*g, x, y];
    );
); }
v=vector(ct, n, v[n]);
v=vecsort(v,1); /* sort according to x + y * g */
v215344=vector(#v,n,v[n][2])
v215345=vector(#v,n,v[n][3])
/* Joerg Arndt, Aug 17 2012 */

cp's OEIS Frontend

User: Peter G. Anderson

A257962 List of successive permutations of the numbers [0,F(n)) achieved by writing the numbers in the Fibonacci/Zeckendorf radix in the form of an (n-2)-bit string, reversing that string, and extracting the numerical value according to Fibonacci/Zeckendorf radix.

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A257961 List of permutations of the intervals of numbers [0,F(n)) defined by x -> x*F(n-1) mod F(n), where F(n) is the n-th Fibonacci number A000045.

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A215781 a(n) = ceiling(n*(sqrt(3)-1)).

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A215592 Value of y in the n-th number of the form x+y*(3+sqrt(13))/2.

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A215591 Value of x in the n-th number of the form x+y*(3+sqrt(13))/2.

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A215590 Value of y in the n-th number of the form x+y*(1+sqrt(2)).

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A215589 Value of x in the n-th number of the form x+y*(1+sqrt(2)).

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A215345 Value of y in the n-th number of the form x+y*(1+sqrt(5))/2.

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PARI

A215344 Value of x in the n-th number of the form x+y*(1+sqrt(5))/2.

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PARI