122 lines
3.9 KiB
Plaintext
122 lines
3.9 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "683859dd",
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"metadata": {},
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"source": [
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"# [Coin Partitions](https://projecteuler.net/problem=78)\n",
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"\n",
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"SageMath once again makes this pretty trivial."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "48a703c0",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"55374"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from itertools import count\n",
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"\n",
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"for n in count(1):\n",
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" if number_of_partitions(n) % 1000000 == 0:\n",
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" break\n",
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"\n",
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"n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ec850c78",
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"metadata": {},
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"source": [
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"Theoretically, we could create a generating function like we did in [problem 31](https://projecteuler.net/problem=76) or [problem 76](https://projecteuler.net/problem=76) to solve this, and could even use $\\mathbb{Z}_{1000000}$ as our base ring to handle the modulus automagically, but since the answer is pretty large, it would be impractical to construct the function with the required precision.\n",
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"\n",
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"Instead, if you'd like to implement the [partition function](https://w.wiki/EoNj) yourself, Euler's [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem) leads to a useful recurrence equation:\n",
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"$$p(n) = p(n - 1) + p(n - 2) - p(n - 5) - p(n - 7) + p(n - 12) + p(n - 15) - p(n - 22) - p(n - 26) + \\cdots$$\n",
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"Here, the numbers are the [generalized pentagonal numbers](https://en.wikipedia.org/wiki/Pentagonal_number). As base cases, $p(0) = 1$ and $p(n) = 0$ for negative $n$, so the infinite series eventually converges.\n",
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"\n",
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"We can directly translate this equation into an implementation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "c8dd39e8",
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"metadata": {},
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"outputs": [],
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"source": [
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"from functools import cache\n",
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"\n",
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"@cache\n",
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"def p(n, modulus):\n",
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" if n < 0:\n",
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" return 0\n",
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" \n",
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" if n == 0:\n",
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" return 1\n",
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" \n",
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" total = 0\n",
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" limit = floor((1 + sqrt(1+24*n)) / 6)\n",
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" \n",
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" # we reverse the range so smaller p(n) values are calculated first and cached\n",
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" # this helps avoid hitting Python's maximum recursion depth\n",
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" for k in reversed(range(1, limit + 1)):\n",
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" total += (-1)^(k+1) * (p(n - polygonal_number(5, k), modulus) + p(n - polygonal_number(5, -k), modulus))\n",
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" total %= modulus\n",
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" \n",
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" return total"
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]
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},
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{
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"cell_type": "markdown",
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"id": "df91171b",
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"metadata": {},
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"source": [
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"Note that SageMath uses libraries like [FLINT](https://flintlib.org/doc/partitions.html) with faster - but much more technical - methods, like the Hardy-Ramanujan-Rademacher formula. This implementation is much easier to both understand and write, but is less performant.\n",
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"\n",
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"## Relevant sequences\n",
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"* Partition numbers: [A000041](https://oeis.org/A000041)\n",
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"* Generalized pentagonal numbers: [A001318](https://oeis.org/A001318)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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