53 lines
2.1 KiB
Markdown
53 lines
2.1 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, un = n3: 1, 8, 27, 64, 125, 216, ...
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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 nth 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) = u1.
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Hence we obtain the following OPs for the cubic sequence:
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OP(1, n) = 1 1, 1, 1, 1, ... OP(2, n) = 7n−6 1, 8, 15, ... OP(3, n) = 6n2−11n+6 1, 8, 27, 58, ... OP(4, n) = n3 1, 8, 27, 64, 125, ...
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Clearly no BOPs exist for k ≥ 4. By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74. Consider the following tenth degree polynomial generating function: un = 1 − n + n2 − n3 + n4 − n5 + n6 − n7 + n8 − n9 + n10 Find the sum of FITs for the BOPs.
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# --hints--
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`euler101()` should return 37076114526.
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```js
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assert.strictEqual(euler101(), 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 euler101() {
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return true;
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}
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euler101();
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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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