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   "source": [
    "# [Counting Summations](https://projecteuler.net/problem=76)\n",
    "\n",
    "We want $p(100) - 1$, where $p(n)$ is the [partition function](https://w.wiki/EoNj). 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."
   ]
  },
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   "source": [
    "number_of_partitions(100) - 1"
   ]
  },
  {
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   "id": "d06d7ff2",
   "metadata": {},
   "source": [
    "Alternatively, if we think of this problem like [problem 31](https://projecteuler.net/problem=31) - just with coins of every possible denomination instead of only a few - we can adapt any of our approaches to that problem, where we construct a generating function."
   ]
  },
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   "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)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "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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