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} \left\lfloor \lfloor m\varphi \rfloor \varphi \right\rfloor & \text{ if } n=1\\ \left\lfloor \lfloor m\varphi \rfloor \varphi^2 \right\rfloor & \text{ if } n=2\\ A_{m,n-2}+A_{m,n-1} & \text{ if }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 | 108240 29890 | 637 It might be fun to know that the largest \$a(n)\$ for \$1\le n\le32767\$ is \$a(32642) = 512653048485188394162163283930413917147479973138989971 = F(256) \lfloor 2 \varphi\rfloor + F(255).\$
Rules
- Input and output are integers
- Your program should at least support input 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
999not9999\$\endgroup\$