add problem 76

notebooks/problem0076.ipynb

@@ -0,0 +1,98 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "c09827d4",
+   "metadata": {},
+   "source": [
+    "# [Counting Summations](https://projecteuler.net/problem=76)\n",
+    "\n",
+    "We want $p(100) - 1$, where $p(n)$ is the [partition function](https://en.wikipedia.org/wiki/Partition_function_(number_theory)). We subtract 1 because $p(n)$ counts $n$ by itself as a partition of $n$, but we only want partitions composed of two or more numbers.\n",
+    "\n",
+    "Guess what? SageMath has this built-in."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "1e4ff844",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "190569291"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "number_of_partitions(100) - 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d06d7ff2",
+   "metadata": {},
+   "source": [
+    "Alternatively, we can adapt any of our approaches to [problem 31](https://projecteuler.net/problem=31), where we construct the generating function. Here's the last approach from that problem."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "bba7ef34",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "190569291"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "R.<x> = PowerSeriesRing(ZZ, default_prec=101)\n",
+    "G = 1 / prod(1 - x^n for n in range(1, 100))\n",
+    "G[100]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "681d7716",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Partition numbers: [A000041](https://oeis.org/A000041)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}