Algorithms - JavaScript - 基礎概念 - 数学的な基礎(帰納法、Fibonacci 数と黄金比、大小、等式)

The Art of Computer Programming
Volume 1
原著

開発環境

• macOS Sierra - Apple (OS)
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• Node.js, Safari(JavaScript エンジン)
• Learning JavaScript [邦訳](参考書籍)

The Art of Computer Programming　Volume 1 Fundamental Algorithms Third Edition 日本語版(Donald E. Knuth (著)、青木 孝 (著)、筧 一彦 (著)、鈴木 健一 (著)、長尾 高弘 (著)、有澤 誠 (その他)、和田 英一 (その他)、ドワンゴ)の第1章(基礎概念)、1.2(数学的な基礎)、演習問題4を取り組んでみる。

4. 
   $\begin{array}{l}\varphi =\frac{\sqrt{5}+1}{2}\\ F_1=1\ge \frac{2}{\sqrt{5}+1}={\varphi }^{-1}={\varphi }^{1-2}\\ F_2=1\ge 1={\varphi }^0={\varphi }^{2-2}\\ \\ F_{n+1}=F_{n-1}+F_n\\ \ge {\varphi }^{n-3}+{\varphi }^{n-2}\\ ={\varphi }^{n-3}\left(1+\varphi \right)\\ ={\varphi }^{n-3}·{\varphi }^2\\ ={\varphi }^{n-1}\\ ={\varphi }^{\left(n+1\right)-2}\end{array}$

コード(Emacs)

HTML5

n = <input id="n0" type="number" min="0" step="1" value="100">
<br>
<button id="run0">run</button>
<button id="clear0">clear</button>
<pre id="output0"></pre>

<script src="sample4.js"></script>

JavaScript

let input0 = document.querySelector('#n0'),
    btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    pre0 = document.querySelector('#output0'),
    p = (x) => pre0.textContent += x + '\n';

let goldenRatio = (1 + Math.sqrt(5)) / 2;

let fibnacci = (n) => {
    let a = 0;
    let b = 1;
    for (i = 2; i <= n; i += 1) {
        b = a + b;
        a = b - a;
    }
    return b;
};

let output = () => {
    let n = parseInt(input0.value, 10);

    p(`n, φ^{n-2}, fibonacci(n), φ^{n-1}, φ^{n-2} <= fibonacci(n) <= φ^{n-1}`);
    for (let i = 0; i <= n; i += 1) {
        let x = Math.pow(goldenRatio, i - 2),
            y = fibnacci(i),
            z = Math.pow(goldenRatio, i - 1);
        p(`${i}, ${x}, ${y}, ${z}: ${x <= y && y <= z}`);
    }
};
    

btn0.onclick = output;
btn1.onclick = () => {
    pre0.textContent = '';
};

output();

n =