Euler 076 - Counting summations.ipynb

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    "Euler Problem 76\n",
    "================\n",
    "\n",
    "It is possible to write five as a sum in exactly six different ways:\n",
    "\n",
    "    4 + 1\n",
    "    3 + 2\n",
    "    3 + 1 + 1\n",
    "    2 + 2 + 1\n",
    "    2 + 1 + 1 + 1\n",
    "    1 + 1 + 1 + 1 + 1\n",
    "\n",
    "How many different ways can one hundred be written as a sum of at least two positive integers?"
   ]
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     "text": [
      "190569292\n"
     ]
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   "source": [
    "# Generate pentagonal numbers\n",
    "# p(k) = k * (3k-1) / 2, k not zero.\n",
    "\n",
    "def pentagonal(N):\n",
    "    a, b = 1, 2\n",
    "    delta = 4\n",
    "    sgn = 1\n",
    "    while a <= N:\n",
    "        yield sgn, a\n",
    "        a += delta\n",
    "        if b <= N:\n",
    "            yield sgn, b\n",
    "            b += delta + 1\n",
    "        delta += 3\n",
    "        sgn = -sgn\n",
    "        \n",
    "N = 100\n",
    "partitions = [0] * (N+1)\n",
    "partitions[0] = 1\n",
    "for n in range(1, N+1):\n",
    "    for sgn, g in pentagonal(n):\n",
    "        partitions[n] += sgn  * partitions[n - g]\n",
    "print(partitions[N])"
   ]
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   "source": [
    "**Explanation:** Uses the pentagonal number recurrence for the partition counting function.\n",
    "\n",
    "http://en.wikipedia.org/wiki/Pentagonal_number_theorem#Partition_recurrence"
   ]
  }
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