New Order #5: where Fibonacci and Beatty meet at Wythoff
$begingroup$
Introduction (may be ignored)
Putting all positive numbers in its regular order (1, 2, 3, ...) is a bit boring, isn't it? So here is a series of challenges around permutations (reshuffelings) of all positive numbers. This is the fifth challenge in this series (links to the first, second, third and fourth challenge).
In this challenge, we will meet the Wythoff array, which is a intertwined avalanche of Fibonacci sequences and Beatty sequences!
The Fibonacci numbers are probably for most of you a well known sequence. Given two starting numbers $F_0$ and $F_1$, the following $F_n$ are given by: $F_n = F_{(n-1)} + F_{(n-2)}$ for $n>2$.
The Beatty sequence, given a parameter $r$ is: $B^r_n = lfloor rn rfloor$ for $n ge 1$. One of the properties of the Beatty sequence is that for every parameter $r$, there is exactly one parameter $s=r/(r-1)$, such that the Beatty sequences for those parameters are disjunct and joined together, they span all natural numbers excluding 0 (e.g.: $B^r cup B^{r/(r-1)} = Bbb{N} setminus {0}$).
Now here comes the mindblowing part: you can create an array, where each row is a Fibonacci sequence and each column is a Beatty sequence. This array is the Wythoff array. The best part is: every positive number appears exactly once in this array! The array looks like this:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
25 41 66 107 173 280 453 733 1186 1919 3105 ...
27 44 71 115 186 301 487 788 1275 2063 3338 ...
...
An element at row $m$ and column $n$ is defined as:
$A_{m,n} = begin{cases}
A_{m,1} = leftlfloor lfloor mvarphi rfloor varphi rightrfloor\
A_{m,2} = leftlfloor lfloor mvarphi rfloor varphi^2 rightrfloor\
A_{m,n} = A_{m,n-2}+A_{m,n-1} text{ for }n > 2
end{cases}$
where $varphi$ is the golden ratio: $varphi=frac{1+sqrt{5}}{2}$.
If we follow the anti-diagonals of this array, we get A035513, which is the target sequence for this challenge (note that this sequence is added to the OEIS by Neil Sloane himself!). Since this is a "pure sequence" challenge, the task is to output $a(n)$ for a given $n$ as input, where $a(n)$ is A035513.
There are different strategies you can follow to get to $a(n)$, which makes this challenge (in my opinion) really interesting.
Task
Given an integer input $n$, output $a(n)$ in integer format, where $a(n)$ is A035513.
Note: 1-based indexing is assumed here; you may use 0-based indexing, so $a(0) = 1; a(1) = 2$, etc. Please mention this in your answer if you choose to use this.
Test cases
Input | Output
---------------
1 | 1
5 | 7
20 | 20
50 | 136
78 | 30
123 | 3194
1234 | 8212236486
3000 | 814
9999 | 740496902
29890 | 637
Rules
- Input and output are integers (your program should at least support input and output in the range of 1 up to 32767). Note that $a(n)$ goes up to 30 digit numbers in this range...
- Invalid input (0, floats, strings, negative values, etc.) may lead to unpredicted output, errors or (un)defined behaviour.
- Default I/O rules apply.
Default loopholes are forbidden.- This is code-golf, so the shortest answers in bytes wins
code-golf sequence
asked 2 hours ago
agtoeveragtoever
1,359424
$endgroup$
|
show 3 more comments
$begingroup$
Introduction (may be ignored)
Putting all positive numbers in its regular order (1, 2, 3, ...) is a bit boring, isn't it? So here is a series of challenges around permutations (reshuffelings) of all positive numbers. This is the fifth challenge in this series (links to the first, second, third and fourth challenge).
In this challenge, we will meet the Wythoff array, which is a intertwined avalanche of Fibonacci sequences and Beatty sequences!
The Fibonacci numbers are probably for most of you a well known sequence. Given two starting numbers $F_0$ and $F_1$, the following $F_n$ are given by: $F_n = F_{(n-1)} + F_{(n-2)}$ for $n>2$.
The Beatty sequence, given a parameter $r$ is: $B^r_n = lfloor rn rfloor$ for $n ge 1$. One of the properties of the Beatty sequence is that for every parameter $r$, there is exactly one parameter $s=r/(r-1)$, such that the Beatty sequences for those parameters are disjunct and joined together, they span all natural numbers excluding 0 (e.g.: $B^r cup B^{r/(r-1)} = Bbb{N} setminus {0}$).
Now here comes the mindblowing part: you can create an array, where each row is a Fibonacci sequence and each column is a Beatty sequence. This array is the Wythoff array. The best part is: every positive number appears exactly once in this array! The array looks like this:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
25 41 66 107 173 280 453 733 1186 1919 3105 ...
27 44 71 115 186 301 487 788 1275 2063 3338 ...
...
An element at row $m$ and column $n$ is defined as:
$A_{m,n} = begin{cases}
A_{m,1} = leftlfloor lfloor mvarphi rfloor varphi rightrfloor\
A_{m,2} = leftlfloor lfloor mvarphi rfloor varphi^2 rightrfloor\
A_{m,n} = A_{m,n-2}+A_{m,n-1} text{ for }n > 2
end{cases}$
where $varphi$ is the golden ratio: $varphi=frac{1+sqrt{5}}{2}$.
If we follow the anti-diagonals of this array, we get A035513, which is the target sequence for this challenge (note that this sequence is added to the OEIS by Neil Sloane himself!). Since this is a "pure sequence" challenge, the task is to output $a(n)$ for a given $n$ as input, where $a(n)$ is A035513.
There are different strategies you can follow to get to $a(n)$, which makes this challenge (in my opinion) really interesting.
Task
Given an integer input $n$, output $a(n)$ in integer format, where $a(n)$ is A035513.
Note: 1-based indexing is assumed here; you may use 0-based indexing, so $a(0) = 1; a(1) = 2$, etc. Please mention this in your answer if you choose to use this.
Test cases
Input | Output
---------------
1 | 1
5 | 7
20 | 20
50 | 136
78 | 30
123 | 3194
1234 | 8212236486
3000 | 814
9999 | 740496902
29890 | 637
Rules
- Input and output are integers (your program should at least support input and output in the range of 1 up to 32767). Note that $a(n)$ goes up to 30 digit numbers in this range...
- Invalid input (0, floats, strings, negative values, etc.) may lead to unpredicted output, errors or (un)defined behaviour.
- Default I/O rules apply.
Default loopholes are forbidden.- This is code-golf, so the shortest answers in bytes wins
code-golf sequence
asked 2 hours ago
agtoeveragtoever
1,359424
$endgroup$
$begingroup$
So what's the New Order reference here?
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
$endgroup$
– agtoever
2 hours ago
$begingroup$
Ah, I completely missed that! Now I feel regret...
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
$endgroup$
– Jonathan Allan
2 hours ago
$begingroup$
@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
$endgroup$
– agtoever
2 hours ago
|
show 3 more comments
$begingroup$
Introduction (may be ignored)
Putting all positive numbers in its regular order (1, 2, 3, ...) is a bit boring, isn't it? So here is a series of challenges around permutations (reshuffelings) of all positive numbers. This is the fifth challenge in this series (links to the first, second, third and fourth challenge).
In this challenge, we will meet the Wythoff array, which is a intertwined avalanche of Fibonacci sequences and Beatty sequences!
The Fibonacci numbers are probably for most of you a well known sequence. Given two starting numbers $F_0$ and $F_1$, the following $F_n$ are given by: $F_n = F_{(n-1)} + F_{(n-2)}$ for $n>2$.
The Beatty sequence, given a parameter $r$ is: $B^r_n = lfloor rn rfloor$ for $n ge 1$. One of the properties of the Beatty sequence is that for every parameter $r$, there is exactly one parameter $s=r/(r-1)$, such that the Beatty sequences for those parameters are disjunct and joined together, they span all natural numbers excluding 0 (e.g.: $B^r cup B^{r/(r-1)} = Bbb{N} setminus {0}$).
Now here comes the mindblowing part: you can create an array, where each row is a Fibonacci sequence and each column is a Beatty sequence. This array is the Wythoff array. The best part is: every positive number appears exactly once in this array! The array looks like this:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
25 41 66 107 173 280 453 733 1186 1919 3105 ...
27 44 71 115 186 301 487 788 1275 2063 3338 ...
...
An element at row $m$ and column $n$ is defined as:
$A_{m,n} = begin{cases}
A_{m,1} = leftlfloor lfloor mvarphi rfloor varphi rightrfloor\
A_{m,2} = leftlfloor lfloor mvarphi rfloor varphi^2 rightrfloor\
A_{m,n} = A_{m,n-2}+A_{m,n-1} text{ for }n > 2
end{cases}$
where $varphi$ is the golden ratio: $varphi=frac{1+sqrt{5}}{2}$.
If we follow the anti-diagonals of this array, we get A035513, which is the target sequence for this challenge (note that this sequence is added to the OEIS by Neil Sloane himself!). Since this is a "pure sequence" challenge, the task is to output $a(n)$ for a given $n$ as input, where $a(n)$ is A035513.
There are different strategies you can follow to get to $a(n)$, which makes this challenge (in my opinion) really interesting.
Task
Given an integer input $n$, output $a(n)$ in integer format, where $a(n)$ is A035513.
Note: 1-based indexing is assumed here; you may use 0-based indexing, so $a(0) = 1; a(1) = 2$, etc. Please mention this in your answer if you choose to use this.
Test cases
Input | Output
---------------
1 | 1
5 | 7
20 | 20
50 | 136
78 | 30
123 | 3194
1234 | 8212236486
3000 | 814
9999 | 740496902
29890 | 637
Rules
- Input and output are integers (your program should at least support input and output in the range of 1 up to 32767). Note that $a(n)$ goes up to 30 digit numbers in this range...
- Invalid input (0, floats, strings, negative values, etc.) may lead to unpredicted output, errors or (un)defined behaviour.
- Default I/O rules apply.
Default loopholes are forbidden.- This is code-golf, so the shortest answers in bytes wins
code-golf sequence
asked 2 hours ago
agtoeveragtoever
1,359424
$endgroup$
Introduction (may be ignored)
Putting all positive numbers in its regular order (1, 2, 3, ...) is a bit boring, isn't it? So here is a series of challenges around permutations (reshuffelings) of all positive numbers. This is the fifth challenge in this series (links to the first, second, third and fourth challenge).
In this challenge, we will meet the Wythoff array, which is a intertwined avalanche of Fibonacci sequences and Beatty sequences!
The Fibonacci numbers are probably for most of you a well known sequence. Given two starting numbers $F_0$ and $F_1$, the following $F_n$ are given by: $F_n = F_{(n-1)} + F_{(n-2)}$ for $n>2$.
The Beatty sequence, given a parameter $r$ is: $B^r_n = lfloor rn rfloor$ for $n ge 1$. One of the properties of the Beatty sequence is that for every parameter $r$, there is exactly one parameter $s=r/(r-1)$, such that the Beatty sequences for those parameters are disjunct and joined together, they span all natural numbers excluding 0 (e.g.: $B^r cup B^{r/(r-1)} = Bbb{N} setminus {0}$).
Now here comes the mindblowing part: you can create an array, where each row is a Fibonacci sequence and each column is a Beatty sequence. This array is the Wythoff array. The best part is: every positive number appears exactly once in this array! The array looks like this:
1 2 3 5 8 13 21 34 55 89 144 ...
4 7 11 18 29 47 76 123 199 322 521 ...
6 10 16 26 42 68 110 178 288 466 754 ...
9 15 24 39 63 102 165 267 432 699 1131 ...
12 20 32 52 84 136 220 356 576 932 1508 ...
14 23 37 60 97 157 254 411 665 1076 1741 ...
17 28 45 73 118 191 309 500 809 1309 2118 ...
19 31 50 81 131 212 343 555 898 1453 2351 ...
22 36 58 94 152 246 398 644 1042 1686 2728 ...
25 41 66 107 173 280 453 733 1186 1919 3105 ...
27 44 71 115 186 301 487 788 1275 2063 3338 ...
...
An element at row $m$ and column $n$ is defined as:
$A_{m,n} = begin{cases}
A_{m,1} = leftlfloor lfloor mvarphi rfloor varphi rightrfloor\
A_{m,2} = leftlfloor lfloor mvarphi rfloor varphi^2 rightrfloor\
A_{m,n} = A_{m,n-2}+A_{m,n-1} text{ for }n > 2
end{cases}$
where $varphi$ is the golden ratio: $varphi=frac{1+sqrt{5}}{2}$.
If we follow the anti-diagonals of this array, we get A035513, which is the target sequence for this challenge (note that this sequence is added to the OEIS by Neil Sloane himself!). Since this is a "pure sequence" challenge, the task is to output $a(n)$ for a given $n$ as input, where $a(n)$ is A035513.
There are different strategies you can follow to get to $a(n)$, which makes this challenge (in my opinion) really interesting.
Task
Given an integer input $n$, output $a(n)$ in integer format, where $a(n)$ is A035513.
Note: 1-based indexing is assumed here; you may use 0-based indexing, so $a(0) = 1; a(1) = 2$, etc. Please mention this in your answer if you choose to use this.
Test cases
Input | Output
---------------
1 | 1
5 | 7
20 | 20
50 | 136
78 | 30
123 | 3194
1234 | 8212236486
3000 | 814
9999 | 740496902
29890 | 637
Rules
- Input and output are integers (your program should at least support input and output in the range of 1 up to 32767). Note that $a(n)$ goes up to 30 digit numbers in this range...
- Invalid input (0, floats, strings, negative values, etc.) may lead to unpredicted output, errors or (un)defined behaviour.
- Default I/O rules apply.
Default loopholes are forbidden.- This is code-golf, so the shortest answers in bytes wins
code-golf sequence
code-golf sequence
asked 2 hours ago
agtoeveragtoever
1,359424
asked 2 hours ago
agtoeveragtoever
1,359424
asked 2 hours ago
agtoeveragtoever
1,359424
asked 2 hours ago
agtoeveragtoever
1,359424
asked 2 hours ago
agtoeveragtoever
1,359424
1,359424
$begingroup$
So what's the New Order reference here?
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
$endgroup$
– agtoever
2 hours ago
$begingroup$
Ah, I completely missed that! Now I feel regret...
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
$endgroup$
– Jonathan Allan
2 hours ago
$begingroup$
@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
$endgroup$
– agtoever
2 hours ago
|
show 3 more comments
$begingroup$
So what's the New Order reference here?
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
$endgroup$
– agtoever
2 hours ago
$begingroup$
Ah, I completely missed that! Now I feel regret...
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
$endgroup$
– Jonathan Allan
2 hours ago
$begingroup$
@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
$endgroup$
– agtoever
2 hours ago
$begingroup$
So what's the New Order reference here?
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
So what's the New Order reference here?
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
$endgroup$
– agtoever
2 hours ago
$begingroup$
@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
$endgroup$
– agtoever
2 hours ago
$begingroup$
Ah, I completely missed that! Now I feel regret...
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
Ah, I completely missed that! Now I feel regret...
$endgroup$
– Luis Mendo
2 hours ago
$begingroup$
Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
$endgroup$
– Jonathan Allan
2 hours ago
$begingroup$
Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
$endgroup$
– Jonathan Allan
2 hours ago
$begingroup$
@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
$endgroup$
– agtoever
2 hours ago
$begingroup$
@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
$endgroup$
– agtoever
2 hours ago
|
show 3 more comments
3 Answers
3
active
oldest
votes
$begingroup$
Jelly, 30 bytes
p`SÞ⁸ịð;Øp,²;¤×Ḟ¥/;+ƝQƊ⁹¡ị@ð/
Try it online!
This is a little slow, but a huge improvement is made with a prefix of Ḥ½Ċ (double, square-root, ceiling) like in this test-suite.
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
$endgroup$
1
$begingroup$
you are right!740496902is the result for999
$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
add a comment |
$begingroup$
Jelly, 27 24 bytes
p`SÞ⁸ịð’;×ØpḞ¥×⁹r‘ÆḞ¤Sð/
Try it online!
Monadic link using 1-based indexing.
Thanks to @JonathanAllan for a better way of getting the row and columns from n and saving 3 bytes. In its shortest form it’s too slow for larger n on TIO, so the following Try it online! reduces the size of the initial list of rows and columns at the cost of three bytes.
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
$endgroup$
$begingroup$
...×⁹r‘ÆḞ¤Sð/saves one in your amalgamation version (TIO)
$endgroup$
– Jonathan Allan
9 mins ago
add a comment |
$begingroup$
Wolfram Language (Mathematica), 90 bytes
Flatten[Table[(F=Fibonacci)[a+1]⌊(b-a+1)GoldenRatio⌋+(b-a)F@a,{b,#},{a,b,1,-1}]][[#]]&
Try it online!
answered 1 hour ago
J42161217J42161217
14k21353
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Jelly, 30 bytes
p`SÞ⁸ịð;Øp,²;¤×Ḟ¥/;+ƝQƊ⁹¡ị@ð/
Try it online!
This is a little slow, but a huge improvement is made with a prefix of Ḥ½Ċ (double, square-root, ceiling) like in this test-suite.
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
$endgroup$
1
$begingroup$
you are right!740496902is the result for999
$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
add a comment |
$begingroup$
Jelly, 30 bytes
p`SÞ⁸ịð;Øp,²;¤×Ḟ¥/;+ƝQƊ⁹¡ị@ð/
Try it online!
This is a little slow, but a huge improvement is made with a prefix of Ḥ½Ċ (double, square-root, ceiling) like in this test-suite.
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
$endgroup$
1
$begingroup$
you are right!740496902is the result for999
$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
add a comment |
$begingroup$
Jelly, 30 bytes
p`SÞ⁸ịð;Øp,²;¤×Ḟ¥/;+ƝQƊ⁹¡ị@ð/
Try it online!
This is a little slow, but a huge improvement is made with a prefix of Ḥ½Ċ (double, square-root, ceiling) like in this test-suite.
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
$endgroup$
Jelly, 30 bytes
p`SÞ⁸ịð;Øp,²;¤×Ḟ¥/;+ƝQƊ⁹¡ị@ð/
Try it online!
This is a little slow, but a huge improvement is made with a prefix of Ḥ½Ċ (double, square-root, ceiling) like in this test-suite.
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
edited 25 mins ago
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
answered 52 mins ago
Jonathan AllanJonathan Allan
54.3k537174
54.3k537174
1
$begingroup$
you are right!740496902is the result for999
$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
add a comment |
1
$begingroup$
you are right!740496902is the result for999
$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
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@NickKennedy - nice, go for it!
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– Jonathan Allan
35 mins ago
1
1
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you are right!
740496902 is the result for 999$endgroup$
– J42161217
47 mins ago
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you are right!
740496902 is the result for 999$endgroup$
– J42161217
47 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
Combining the first part of yours and second part of mine gives 25 bytes. Not sure which of us should have the combined version!
$endgroup$
– Nick Kennedy
37 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
$begingroup$
@NickKennedy - nice, go for it!
$endgroup$
– Jonathan Allan
35 mins ago
add a comment |
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Jelly, 27 24 bytes
p`SÞ⁸ịð’;×ØpḞ¥×⁹r‘ÆḞ¤Sð/
Try it online!
Monadic link using 1-based indexing.
Thanks to @JonathanAllan for a better way of getting the row and columns from n and saving 3 bytes. In its shortest form it’s too slow for larger n on TIO, so the following Try it online! reduces the size of the initial list of rows and columns at the cost of three bytes.
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
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$begingroup$
...×⁹r‘ÆḞ¤Sð/saves one in your amalgamation version (TIO)
$endgroup$
– Jonathan Allan
9 mins ago
add a comment |
$begingroup$
Jelly, 27 24 bytes
p`SÞ⁸ịð’;×ØpḞ¥×⁹r‘ÆḞ¤Sð/
Try it online!
Monadic link using 1-based indexing.
Thanks to @JonathanAllan for a better way of getting the row and columns from n and saving 3 bytes. In its shortest form it’s too slow for larger n on TIO, so the following Try it online! reduces the size of the initial list of rows and columns at the cost of three bytes.
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
$endgroup$
$begingroup$
...×⁹r‘ÆḞ¤Sð/saves one in your amalgamation version (TIO)
$endgroup$
– Jonathan Allan
9 mins ago
add a comment |
$begingroup$
Jelly, 27 24 bytes
p`SÞ⁸ịð’;×ØpḞ¥×⁹r‘ÆḞ¤Sð/
Try it online!
Monadic link using 1-based indexing.
Thanks to @JonathanAllan for a better way of getting the row and columns from n and saving 3 bytes. In its shortest form it’s too slow for larger n on TIO, so the following Try it online! reduces the size of the initial list of rows and columns at the cost of three bytes.
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
$endgroup$
Jelly, 27 24 bytes
p`SÞ⁸ịð’;×ØpḞ¥×⁹r‘ÆḞ¤Sð/
Try it online!
Monadic link using 1-based indexing.
Thanks to @JonathanAllan for a better way of getting the row and columns from n and saving 3 bytes. In its shortest form it’s too slow for larger n on TIO, so the following Try it online! reduces the size of the initial list of rows and columns at the cost of three bytes.
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
edited 2 mins ago
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
answered 48 mins ago
Nick KennedyNick Kennedy
1,56649
1,56649
$begingroup$
...×⁹r‘ÆḞ¤Sð/saves one in your amalgamation version (TIO)
$endgroup$
– Jonathan Allan
9 mins ago
add a comment |
$begingroup$
...×⁹r‘ÆḞ¤Sð/saves one in your amalgamation version (TIO)
$endgroup$
– Jonathan Allan
9 mins ago
$begingroup$
...×⁹r‘ÆḞ¤Sð/ saves one in your amalgamation version (TIO)$endgroup$
– Jonathan Allan
9 mins ago
$begingroup$
...×⁹r‘ÆḞ¤Sð/ saves one in your amalgamation version (TIO)$endgroup$
– Jonathan Allan
9 mins ago
add a comment |
$begingroup$
Wolfram Language (Mathematica), 90 bytes
Flatten[Table[(F=Fibonacci)[a+1]⌊(b-a+1)GoldenRatio⌋+(b-a)F@a,{b,#},{a,b,1,-1}]][[#]]&
Try it online!
answered 1 hour ago
J42161217J42161217
14k21353
$endgroup$
add a comment |
$begingroup$
Wolfram Language (Mathematica), 90 bytes
Flatten[Table[(F=Fibonacci)[a+1]⌊(b-a+1)GoldenRatio⌋+(b-a)F@a,{b,#},{a,b,1,-1}]][[#]]&
Try it online!
answered 1 hour ago
J42161217J42161217
14k21353
$endgroup$
add a comment |
$begingroup$
Wolfram Language (Mathematica), 90 bytes
Flatten[Table[(F=Fibonacci)[a+1]⌊(b-a+1)GoldenRatio⌋+(b-a)F@a,{b,#},{a,b,1,-1}]][[#]]&
Try it online!
answered 1 hour ago
J42161217J42161217
14k21353
$endgroup$
Wolfram Language (Mathematica), 90 bytes
Flatten[Table[(F=Fibonacci)[a+1]⌊(b-a+1)GoldenRatio⌋+(b-a)F@a,{b,#},{a,b,1,-1}]][[#]]&
Try it online!
answered 1 hour ago
J42161217J42161217
14k21353
answered 1 hour ago
J42161217J42161217
14k21353
answered 1 hour ago
J42161217J42161217
14k21353
answered 1 hour ago
J42161217J42161217
14k21353
14k21353
add a comment |
add a comment |
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$begingroup$
So what's the New Order reference here?
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– Luis Mendo
2 hours ago
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@LuisMendo: the avalanche of Fibonacci and Beatty sequences, which form the Wythoff array...
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– agtoever
2 hours ago
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Ah, I completely missed that! Now I feel regret...
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– Luis Mendo
2 hours ago
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Is a floating point representation of phi (or rt(5)) and application of the recurrence going to satisfy the range requirement?
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– Jonathan Allan
2 hours ago
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@JonathanAllan : good point... I'll look into that later. For now: let's pose that if some code passes the test cases, then it works sufficiently.
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– agtoever
2 hours ago