2018-09-30 23:01:58 +01:00
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---
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id: 5900f5201000cf542c510032
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title: 'Problem 435: Polynomials of Fibonacci numbers'
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2020-11-27 19:02:05 +01:00
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challengeType: 5
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2019-08-05 09:17:33 -07:00
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forumTopicId: 302106
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2021-01-13 03:31:00 +01:00
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dashedName: problem-435-polynomials-of-fibonacci-numbers
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2018-09-30 23:01:58 +01:00
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---
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2020-11-27 19:02:05 +01:00
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# --description--
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2021-07-29 20:14:09 +02:00
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The Fibonacci numbers $\\{f_n, n ≥ 0\\}$ are defined recursively as $f_n = f_{n - 1} + f_{n - 2}$ with base cases $f_0 = 0$ and $f_1 = 1$.
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2020-11-27 19:02:05 +01:00
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2021-07-29 20:14:09 +02:00
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Define the polynomials $\\{F_n, n ≥ 0\\}$ as $F_n(x) = \displaystyle\sum_{i = 0}^n f_ix^i$.
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2018-09-30 23:01:58 +01:00
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2021-07-29 20:14:09 +02:00
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For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\\,357\\,683$.
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2018-09-30 23:01:58 +01:00
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2021-07-29 20:14:09 +02:00
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Let $n = {10}^{15}$. Find the sum $\displaystyle\sum_{x = 0}^{100} F_n(x)$ and give your answer modulo $1\\,307\\,674\\,368\\,000 \\, (= 15!)$.
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2018-09-30 23:01:58 +01:00
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2020-11-27 19:02:05 +01:00
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# --hints--
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2018-09-30 23:01:58 +01:00
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2021-07-29 20:14:09 +02:00
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`polynomialsOfFibonacciNumbers()` should return `252541322550`.
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2018-09-30 23:01:58 +01:00
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2020-11-27 19:02:05 +01:00
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```js
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2021-07-29 20:14:09 +02:00
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assert.strictEqual(polynomialsOfFibonacciNumbers(), 252541322550);
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2018-09-30 23:01:58 +01:00
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```
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2020-11-27 19:02:05 +01:00
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# --seed--
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2018-09-30 23:01:58 +01:00
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2020-11-27 19:02:05 +01:00
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## --seed-contents--
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2018-09-30 23:01:58 +01:00
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```js
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2021-07-29 20:14:09 +02:00
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function polynomialsOfFibonacciNumbers() {
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2020-09-15 09:57:40 -07:00
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2018-09-30 23:01:58 +01:00
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return true;
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}
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2021-07-29 20:14:09 +02:00
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polynomialsOfFibonacciNumbers();
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2018-09-30 23:01:58 +01:00
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```
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2020-11-27 19:02:05 +01:00
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# --solutions--
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2018-09-30 23:01:58 +01:00
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```js
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// solution required
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```
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