57 lines
2.3 KiB
Markdown
57 lines
2.3 KiB
Markdown
---
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id: 5900f3d21000cf542c50fee4
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title: 'Problem 101: Optimum polynomial'
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challengeType: 5
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forumTopicId: 301725
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dashedName: problem-101-optimum-polynomial
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---
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# --description--
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If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
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As an example, let us consider the sequence of cube numbers. This is defined by the generating function, $u_n = n^3: 1, 8, 27, 64, 125, 216, \ldots$
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Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
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We shall define $OP(k, n)$ to be the $n^{th}$ term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that $OP(k, n)$ will accurately generate the terms of the sequence for $n ≤ k$, and potentially the first incorrect term (FIT) will be $OP(k, k+1)$; in which case we shall call it a bad OP (BOP).
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As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for $n ≥ 2, OP(1, n) = u_1$.
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Hence we obtain the following OPs for the cubic sequence:
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$$\begin{array}{ll} OP(1, n) = 1 & 1, {\color{red}1}, 1, 1, \ldots \\\\ OP(2, n) = 7n−6 & 1, 8, {\color{red}{15}}, \ldots \\\\ OP(3, n) = 6n^2−11n+6 & 1, 8, 27, {\color{red}{58}}, \ldots \\\\ OP(4, n) = n^3 & 1, 8, 27, 64, 125, \ldots \end{array}$$
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Clearly no BOPs exist for k ≥ 4. By considering the sum of FITs generated by the BOPs (indicated in $\color{red}{red}$ above), we obtain 1 + 15 + 58 = 74. Consider the following tenth degree polynomial generating function:
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$$u_n = 1 − n + n^2 − n^3 + n^4 − n^5 + n^6 − n^7 + n^8 − n^9 + n^{10}$$
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Find the sum of FITs for the BOPs.
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# --hints--
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`optimumPolynomial()` should return `37076114526`.
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```js
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assert.strictEqual(optimumPolynomial(), 37076114526);
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```
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# --seed--
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## --seed-contents--
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```js
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function optimumPolynomial() {
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return true;
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}
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optimumPolynomial();
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```
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# --solutions--
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```js
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// solution required
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```
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