47 lines
1.8 KiB
Markdown
47 lines
1.8 KiB
Markdown
If we are presented with the first k terms of a
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sequence it is impossible to say with certainty the value of the next term,
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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.
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This is defined by the generating function,
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~~~
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u_n = n^3: 1, 8, 27, 64, 125, 216, ...
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~~~
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Suppose we were only given the first two terms of this sequence.
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Working on the principle that "simple is best" we should assume a linear
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relationship and predict the next term to be 15 (common difference 7).
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Even if we were presented with the first three terms, by the same
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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
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generating function for the first k terms of a sequence.
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It should be clear that `OP(k, n)` will accurately generate the terms
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of the sequence for `n <= k`, and potentially the first incorrect term (FIT)
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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,
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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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~~~
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OP(1, n) = 1 1, **1**, 1, 1, ...
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OP(2, n) = 7n-6 1, 8, **15**, ...
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OP(3, n) = 6n^2-11n+6 1, 8, 27, **58**, ...
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OP(4, n) = n^3 1, 8, 27, 64, 125, ...
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~~~
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Clearly no BOPs exist for `k >= 4`.
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By considering the sum of FITs generated by the BOPs (indicated in **red** above),
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we obtain `1 + 15 + 58 = 74`.
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Consider the following tenth degree polynomial generating function:
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~~~
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un = 1 - n + n^2 - n^3 + n^4 - n^5 + n^6 - n^7 + n^8 - n^9 + n^10
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~~~
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Find the sum of FITs for the BOPs. |