derivation of S_{2x+1}

collatz

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   "id": "5cdc3c96-246e-4750-960e-342c9bd6ac2e",
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   "source": [
    "import sympy as sy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43a65d5a-63fd-4a86-b6bd-c929af36a74d",
   "metadata": {},
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   "source": [
    "S=sy.IndexedBase('S')\n",
    "N,L,x = sy.symbols('N L x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "38786401-53a5-49fc-912b-d954963a3eec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle {S}_{x} = 2^{N} \\left(2 x + 1\\right) - 3^{L} x$"
      ],
      "text/plain": [
       "Eq(S[x], 2**N*(2*x + 1) - 3**L*x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle {S}_{2 x + 1} = 2^{N} \\left(4 x + 2\\right) - 3^{L} \\left(2 x + 1\\right)$"
      ],
      "text/plain": [
       "Eq(S[2*x + 1], 2**N*(4*x + 2) - 3**L*(2*x + 1))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# S[x] = -3L(x) + 2N(2x + 1)\n",
    "eqn1=sy.Eq(S[x], -3**L*x + 2**N*(2*x + 1))\n",
    "# S[2x+1] = -3L(2x + 1) + 2N(4x + 2)\n",
    "eqn2=sy.Eq(S[2*x+1], (-3**L)*(2*x + 1) + (2**N)*(4*x + 2))\n",
    "\n",
    "display(eqn1, eqn2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "d925482f-1e63-42c9-9639-b5a55cd6fdea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x + 1 = 2^{- N} \\left(3^{L} x + {S}_{x}\\right)$"
      ],
      "text/plain": [
       "Eq(2*x + 1, (3**L*x + S[x])/2**N)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x + 1 = 3^{- L} \\left(2^{N + 1} + 2^{N + 2} x - {S}_{2 x + 1}\\right)$"
      ],
      "text/plain": [
       "Eq(2*x + 1, (2**(N + 1) + 2**(N + 2)*x - S[2*x + 1])/3**L)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2^{- N} \\left(3^{L} x + {S}_{x}\\right) = 3^{- L} \\left(2^{N + 1} + 2^{N + 2} x - {S}_{2 x + 1}\\right)$"
      ],
      "text/plain": [
       "Eq((3**L*x + S[x])/2**N, (2**(N + 1) + 2**(N + 2)*x - S[2*x + 1])/3**L)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr1=sy.solve(eqn1, (2*x+1))[0]\n",
    "expr2=sy.solve(eqn2, (2*x+1))[0]\n",
    "eqn3=sy.Eq(expr1, expr2)\n",
    "display(sy.Eq(2*x+1, expr1), sy.Eq(2*x+1, expr2), eqn3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1599d460-2d6c-48a8-a170-7005390a298a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle {S}_{2 x + 1} = 2^{- N} \\left(2^{2 N + 1} \\left(2 x + 1\\right) - 3^{L} {S}_{x} - 9^{L} x\\right)$"
      ],
      "text/plain": [
       "Eq(S[2*x + 1], (2**(2*N + 1)*(2*x + 1) - 3**L*S[x] - 9**L*x)/2**N)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eqn4=sy.Eq(S[2*x+1], sy.solve(eqn3, S[2*x+1])[0])\n",
    "display(eqn4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "707c0fa9-2d5b-4544-8790-2b163f16425b",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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