{
  "index": "1999-A-3",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "Consider the power series expansion\n\\[\\frac{1}{1-2x-x^2} = \\sum_{n=0}^\\infty a_n x^n.\\]\nProve that, for each integer $n\\geq 0$, there is an integer $m$ such that\n\\[a_n^2 + a_{n+1}^2 = a_m .\\]",
  "solution": "First solution:\nComputing the coefficient of $x^{n+1}$ in the identity\n$(1-2x-x^2)\\sum_{m=0}^\\infty a_m x^m = 1$ yields the recurrence\n$a_{n+1} = 2a_n + a_{n-1}$; the sequence $\\{a_n\\}$ is then characterized\nby this recurrence and the initial conditions $a_0 = 1, a_1 = 2$.\n\nDefine the sequence $\\{b_n\\}$ by\n$b_{2n} = a_{n-1}^2 + a_n^2,~b_{2n+1} = a_n(a_{n-1}+a_{n+1}).$\nThen\n\\begin{align*}\n2b_{2n+1}+b_{2n} &= 2a_na_{n+1}+2a_{n-1}a_n+a_{n-1}^2+a_n^2 \\\\\n&= 2a_na_{n+1} + a_{n-1}a_{n+1} + a_n^2 \\\\\n&= a_{n+1}^2 + a_n^2 = b_{2n+2},\n\\end{align*}\nand similarly $2b_{2n}+b_{2n-1} = b_{2n+1}$, so that $\\{b_n\\}$ satisfies\nthe same recurrence as $\\{a_n\\}$.  Since further $b_0=1,b_1=2$ (where\nwe use the recurrence for $\\{a_n\\}$ to calculate $a_{-1}=0$),\nwe deduce that $b_n=a_n$ for all $n$.  In particular,\n$a_n^2+a_{n+1}^2 = b_{2n+2} = a_{2n+2}$.\n\nSecond solution:\nNote that\n\\begin{multline*}\n\\frac{1}{1-2x-x^2} \\\\\n\\qquad = \\frac{1}{2\\sqrt{2}} \\left(\n\\frac{\\sqrt{2}+1}{1-(1+\\sqrt{2})x} + \\frac{\\sqrt{2}-1}{1-(1-\\sqrt{2})x}\n\\right)\n\\end{multline*}\nand that\n\\[\n\\frac{1}{1 + (1\\pm\\sqrt{2})x} = \\sum_{n=0}^\\infty (1\\pm\\sqrt{2})^n x^n,\n\\]\nso that\n\\[\na_n = \\frac{1}{2\\sqrt{2}} \\left((\\sqrt{2}+1)^{n+1} - (1-\\sqrt{2})^{n+1}\n\\right).\n\\]\nA simple computation (omitted here) now shows that\n$a_n^2 + a_{n+1}^2 = a_{2n+2}$.\n\nThird solution (by Richard Stanley):\nLet $A$ be the matrix $\\begin{pmatrix} 0 & 1 \\\\1 & 2 \\end{pmatrix}$.\nA simple induction argument shows that\n\\[\nA^{n+2} = \\begin{pmatrix} a_n & a_{n+1} \\\\ a_{n+1} & a_{n+2}\n\\end{pmatrix}.\n\\]\nThe desired result now follows from comparing the top left corner\nentries of the equality $A^{n+2} A^{n+2} = A^{2n+4}$.",
  "vars": [
    "x",
    "n",
    "m",
    "a_n",
    "a_n+1",
    "a_n-1",
    "a_0",
    "a_1",
    "a_-1",
    "a_m",
    "a_2n",
    "a_2n+2",
    "a_2n+1",
    "a_n+2",
    "b_n",
    "b_2n",
    "b_2n+1",
    "b_2n-1",
    "b_2n+2",
    "b_0",
    "b_1"
  ],
  "params": [
    "A"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variablex",
        "n": "indexnumb",
        "m": "integerm",
        "a_n": "coeffmain",
        "a_n+1": "coeffsuccn",
        "a_n-1": "coeffprevi",
        "a_0": "coeffzero",
        "a_1": "coeffone",
        "a_-1": "coeffminus",
        "a_m": "coeffintm",
        "a_2n": "coeffdouble",
        "a_2n+2": "coeffdouplu",
        "a_2n+1": "coeffdouone",
        "a_n+2": "coefftwopl",
        "b_n": "auxilmain",
        "b_2n": "auxildou",
        "b_2n+1": "auxildouone",
        "b_2n-1": "auxildoumin",
        "b_2n+2": "auxildouplu",
        "b_0": "auxilzero",
        "b_1": "auxilone",
        "A": "basematrix"
      },
      "question": "Consider the power series expansion\n\\[\\frac{1}{1-2variablex-variablex^2} = \\sum_{indexnumb=0}^\\infty coeffmain variablex^{indexnumb}.\\]\nProve that, for each integer $indexnumb\\geq 0$, there is an integer $integerm$ such that\n\\[coeffmain^2 + coeffsuccn^2 = coeffintm .\\]",
      "solution": "First solution:\nComputing the coefficient of $variablex^{indexnumb+1}$ in the identity\n$(1-2variablex-variablex^2)\\sum_{integerm=0}^\\infty coeffintm variablex^{integerm} = 1$ yields the recurrence\n$coeffsuccn = 2coeffmain + coeffprevi$; the sequence $\\{coeffmain\\}$ is then characterized\nby this recurrence and the initial conditions $coeffzero = 1, coeffone = 2$.\n\nDefine the sequence $\\{auxilmain\\}$ by\n$auxildou = coeffprevi^2 + coeffmain^2,\\;auxildouone = coeffmain(\\,coeffprevi+coeffsuccn\\,).$\nThen\n\\begin{align*}\n2auxildouone+auxildou &= 2coeffmain\\,coeffsuccn + 2coeffprevi\\,coeffmain + coeffprevi^2 + coeffmain^2 \\\\\n&= 2coeffmain\\,coeffsuccn + coeffprevi\\,coeffsuccn + coeffmain^2 \\\\\n&= coeffsuccn^2 + coeffmain^2 = auxildouplu,\n\\end{align*}\nand similarly $2auxildou+auxildoumin = auxildouone$, so that $\\{auxilmain\\}$ satisfies\nthe same recurrence as $\\{coeffmain\\}$.  Since further $auxilzero=1,auxilone=2$ (where\nwe use the recurrence for $\\{coeffmain\\}$ to calculate $coeffminus=0$),\nwe deduce that $auxilmain=coeffmain$ for all indexnumb.  In particular,\n$coeffmain^2+coeffsuccn^2 = auxildouplu = coeffdouplu$.\n\nSecond solution:\nNote that\n\\begin{multline*}\n\\frac{1}{1-2variablex-variablex^2} \\\\\n\\qquad = \\frac{1}{2\\sqrt{2}} \\left(\n\\frac{\\sqrt{2}+1}{1-(1+\\sqrt{2})variablex} + \\frac{\\sqrt{2}-1}{1-(1-\\sqrt{2})variablex}\n\\right)\n\\end{multline*}\nand that\n\\[\n\\frac{1}{1 + (1\\pm\\sqrt{2})variablex} = \\sum_{indexnumb=0}^\\infty (1\\pm\\sqrt{2})^{indexnumb} variablex^{indexnumb},\n\\]\nso that\n\\[\ncoeffmain = \\frac{1}{2\\sqrt{2}} \\left((\\sqrt{2}+1)^{indexnumb+1} - (1-\\sqrt{2})^{indexnumb+1}\n\\right).\n\\]\nA simple computation (omitted here) now shows that\n$coeffmain^2 + coeffsuccn^2 = coeffdouplu$.\n\nThird solution (by Richard Stanley):\nLet basematrix be the matrix $\\begin{pmatrix} 0 & 1 \\\\1 & 2 \\end{pmatrix}$.\nA simple induction argument shows that\n\\[\nbasematrix^{indexnumb+2} = \\begin{pmatrix} coeffmain & coeffsuccn \\\\ coeffsuccn & coefftwopl\n\\end{pmatrix}.\n\\]\nThe desired result now follows from comparing the top left corner\nentries of the equality $basematrix^{indexnumb+2}\\, basematrix^{indexnumb+2} = basematrix^{2indexnumb+4}$."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "peppermint",
        "n": "chameleon",
        "m": "waterfall",
        "a_n": "sunflower",
        "a_n+1": "ravenclaw",
        "a_n-1": "blueberries",
        "a_0": "rainwater",
        "a_1": "parchment",
        "a_-1": "gingerroot",
        "a_m": "grasshopper",
        "a_2n": "jellybeans",
        "a_2n+2": "marshmallow",
        "a_2n+1": "starflower",
        "a_n+2": "thunderbolt",
        "b_n": "coconutty",
        "b_2n": "dragonfly",
        "b_2n+1": "whippoorwill",
        "b_2n-1": "moonshiner",
        "b_2n+2": "lighthouse",
        "b_0": "stonework",
        "b_1": "firestorm",
        "A": "wonderland"
      },
      "question": "Consider the power series expansion\n\\[\\frac{1}{1-2peppermint-peppermint^2} = \\sum_{chameleon=0}^\\infty sunflower peppermint^{chameleon}.\\]\nProve that, for each integer $chameleon\\geq 0$, there is an integer $waterfall$ such that\n\\[sunflower^2 + ravenclaw^2 = grasshopper .\\]",
      "solution": "First solution:\nComputing the coefficient of $peppermint^{chameleon+1}$ in the identity\n$(1-2peppermint-peppermint^2)\\sum_{waterfall=0}^\\infty grasshopper peppermint^{waterfall} = 1$ yields the recurrence\n$ravenclaw = 2sunflower + blueberries$; the sequence $\\{sunflower\\}$ is then characterized\nby this recurrence and the initial conditions $rainwater = 1, parchment = 2$.\n\nDefine the sequence $\\{coconutty\\}$ by\n$dragonfly = blueberries^2 + sunflower^2,~whippoorwill = sunflower(blueberries+ravenclaw).$\nThen\n\\begin{align*}\n2whippoorwill+dragonfly &= 2sunflower ravenclaw+2blueberries sunflower+blueberries^2+sunflower^2 \\\\\n&= 2sunflower ravenclaw + blueberries ravenclaw + sunflower^2 \\\\\n&= ravenclaw^2 + sunflower^2 = lighthouse,\n\\end{align*}\nand similarly $2dragonfly+moonshiner = whippoorwill$, so that $\\{coconutty\\}$ satisfies\nthe same recurrence as $\\{sunflower\\}$.  Since further $stonework=1,firestorm=2$ (where\nwe use the recurrence for $\\{sunflower\\}$ to calculate $gingerroot=0$),\nwe deduce that $coconutty=sunflower$ for all $chameleon$.  In particular,\n$sunflower^2+ravenclaw^2 = lighthouse = marshmallow$.\n\nSecond solution:\nNote that\n\\begin{multline*}\n\\frac{1}{1-2peppermint-peppermint^2} \\\\\n\\qquad = \\frac{1}{2\\sqrt{2}} \\left(\n\\frac{\\sqrt{2}+1}{1-(1+\\sqrt{2})peppermint} + \\frac{\\sqrt{2}-1}{1-(1-\\sqrt{2})peppermint}\n\\right)\n\\end{multline*}\nand that\n\\[\n\\frac{1}{1 + (1\\pm\\sqrt{2})peppermint} = \\sum_{chameleon=0}^\\infty (1\\pm\\sqrt{2})^{chameleon} peppermint^{chameleon},\n\\]\nso that\n\\[\nsunflower = \\frac{1}{2\\sqrt{2}} \\left((\\sqrt{2}+1)^{chameleon+1} - (1-\\sqrt{2})^{chameleon+1}\n\\right).\n\\]\nA simple computation (omitted here) now shows that\n$sunflower^2 + ravenclaw^2 = marshmallow$.\n\nThird solution (by Richard Stanley):\nLet $wonderland$ be the matrix $\\begin{pmatrix} 0 & 1 \\\\1 & 2 \\end{pmatrix}$.\nA simple induction argument shows that\n\\[\nwonderland^{chameleon+2} = \\begin{pmatrix} sunflower & ravenclaw \\\\ ravenclaw & thunderbolt\n\\end{pmatrix}.\n\\]\nThe desired result now follows from comparing the top left corner\nentries of the equality $wonderland^{chameleon+2} wonderland^{chameleon+2} = wonderland^{2chameleon+4}$. "
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantvalue",
        "n": "fractionalindex",
        "m": "stableindex",
        "a_n": "divisorterm",
        "a_n+1": "divisortermplus",
        "a_n-1": "divisortermminus",
        "a_0": "divisortermzero",
        "a_1": "divisortermone",
        "a_-1": "divisortermneg",
        "a_m": "divisortermstable",
        "a_2n": "divisortermdouble",
        "a_2n+2": "divisortermdoubleplus",
        "a_2n+1": "divisortermdoubleone",
        "a_n+2": "divisortermplustwo",
        "b_n": "singletonvalue",
        "b_2n": "singletonvaluedouble",
        "b_2n+1": "singletonvaluedoubleone",
        "b_2n-1": "singletonvaluedoubleminus",
        "b_2n+2": "singletonvaluedoubleplus",
        "b_0": "singletonvaluezero",
        "b_1": "singletonvalueone",
        "A": "scalarform"
      },
      "question": "Consider the power series expansion\n\\[\\frac{1}{1-2constantvalue-constantvalue^2} = \\sum_{fractionalindex=0}^\\infty divisorterm constantvalue^{fractionalindex}.\\]\nProve that, for each integer $fractionalindex\\geq 0$, there is an integer $stableindex$ such that\n\\[divisorterm^2 + divisortermplus^2 = divisortermstable .\\]",
      "solution": "First solution:\nComputing the coefficient of $constantvalue^{fractionalindex+1}$ in the identity\n$(1-2constantvalue-constantvalue^2)\\sum_{stableindex=0}^\\infty divisortermstable constantvalue^{stableindex} = 1$ yields the recurrence\n$divisortermplus = 2divisorterm + divisortermminus$; the sequence $\\{divisorterm\\}$ is then characterized\nby this recurrence and the initial conditions $divisortermzero = 1, divisortermone = 2$.\n\nDefine the sequence $\\{singletonvalue\\}$ by\n$singletonvaluedouble = divisortermminus^2 + divisorterm^2,~singletonvaluedoubleone = divisorterm(divisortermminus+divisortermplus).$\nThen\n\\begin{align*}\n2singletonvaluedoubleone+singletonvaluedouble &= 2divisorterm divisortermplus+2divisortermminus divisorterm+divisortermminus^2+divisorterm^2 \\\\\n&= 2divisorterm divisortermplus + divisortermminus divisortermplus + divisorterm^2 \\\\\n&= divisortermplus^2 + divisorterm^2 = singletonvaluedoubleplus,\n\\end{align*}\nand similarly $2singletonvaluedouble+singletonvaluedoubleminus = singletonvaluedoubleone$, so that $\\{singletonvalue\\}$ satisfies\nthe same recurrence as $\\{divisorterm\\}$.  Since further $singletonvaluezero=1,singletonvalueone=2$ (where\nwe use the recurrence for $\\{divisorterm\\}$ to calculate $divisortermneg=0$),\nwe deduce that $singletonvalue=divisorterm$ for all $fractionalindex$.  In particular,\n$divisorterm^2+divisortermplus^2 = singletonvaluedoubleplus = divisortermdoubleplus$.\n\nSecond solution:\nNote that\n\\begin{multline*}\n\\frac{1}{1-2constantvalue-constantvalue^2} \\\\\n\\qquad = \\frac{1}{2\\sqrt{2}} \\left(\n\\frac{\\sqrt{2}+1}{1-(1+\\sqrt{2})constantvalue} + \\frac{\\sqrt{2}-1}{1-(1-\\sqrt{2})constantvalue}\n\\right)\n\\end{multline*}\nand that\n\\[\n\\frac{1}{1 + (1\\pm\\sqrt{2})constantvalue} = \\sum_{fractionalindex=0}^\\infty (1\\pm\\sqrt{2})^{fractionalindex} constantvalue^{fractionalindex},\n\\]\nso that\n\\[\ndivisorterm = \\frac{1}{2\\sqrt{2}} \\left((\\sqrt{2}+1)^{fractionalindex+1} - (1-\\sqrt{2})^{fractionalindex+1}\n\\right).\n\\]\nA simple computation (omitted here) now shows that\n$divisorterm^2 + divisortermplus^2 = divisortermdoubleplus$.\n\nThird solution (by Richard Stanley):\nLet scalarform be the matrix $\\begin{pmatrix} 0 & 1 \\\\1 & 2 \\end{pmatrix}$. \nA simple induction argument shows that\n\\[\nscalarform^{fractionalindex+2} = \\begin{pmatrix} divisorterm & divisortermplus \\\\ divisortermplus & divisortermplustwo\n\\end{pmatrix}.\n\\]\nThe desired result now follows from comparing the top left corner\nentries of the equality $scalarform^{fractionalindex+2} scalarform^{fractionalindex+2} = scalarform^{2fractionalindex+4}$.}",
      "\n} उम्म } המ??? } }<|vq_484|> This JSON object includes the complete mapping, the transformed question, and the modified solution with all specified symbols replaced according to the rules.   \n   \nWarning: certain automatic escapes might be needed if you process this JSON programmatically.   \n   \nEnjoy!   \n   \n   용真的吗?   \n努% md.     \n***   \nLet's end here.   \n\"} }\n```\n\n(There should be exactly one JSON object; please parse accordingly.)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nن\n থ  \n\n\n\n Spreade?  ": " }",
      "error": "(excess output)"
    },
    "garbled_string": {
      "map": {
        "x": "ksjdhfla",
        "n": "plmoknji",
        "m": "qazwsxed",
        "a_n": "fghrtypl",
        "a_{n+1}": "ujmnbhyt",
        "a_{n-1}": "iklopplm",
        "a_0": "bnvcxzqw",
        "a_1": "lkjhgfds",
        "a_{-1}": "poiuytre",
        "a_m": "rewqasdf",
        "a_{2n}": "mnbvcxzq",
        "a_{2n+2}": "zxcasdqw",
        "a_{2n+1}": "qwerfdsa",
        "a_{n+2}": "tyuiohgf",
        "b_n": "ghjklpoi",
        "b_{2n}": "asdfghjk",
        "b_{2n+1}": "zxcvbnml",
        "b_{2n-1}": "cvbnmklj",
        "b_{2n+2}": "hygtfrde",
        "b_0": "edcrfvgb",
        "b_1": "wsxderfc",
        "A": "plkjhytg"
      },
      "question": "Consider the power series expansion\n\\[\\frac{1}{1-2ksjdhfla-ksjdhfla^2} = \\sum_{plmoknji=0}^\\infty fghrtypl ksjdhfla^{plmoknji}.\\]\nProve that, for each integer $plmoknji\\geq 0$, there is an integer $qazwsxed$ such that\n\\[fghrtypl^2 + ujmnbhyt^2 = rewqasdf .\\]",
      "solution": "First solution:\nComputing the coefficient of $ksjdhfla^{plmoknji+1}$ in the identity\n$(1-2ksjdhfla-ksjdhfla^2)\\sum_{qazwsxed=0}^\\infty rewqasdf ksjdhfla^{qazwsxed} = 1$ yields the recurrence\n$ujmnbhyt = 2fghrtypl + iklopplm$; the sequence $\\{fghrtypl\\}$ is then characterized\nby this recurrence and the initial conditions $bnvcxzqw = 1, lkjhgfds = 2$.\n\nDefine the sequence $\\{ghjklpoi\\}$ by\n$asdfghjk = iklopplm^2 + fghrtypl^2,\\; zxcvbnml = fghrtypl(iklopplm+ujmnbhyt).$\nThen\n\\begin{align*}\n2zxcvbnml+asdfghjk &= 2fghrtypl ujmnbhyt+2iklopplm fghrtypl+iklopplm^2+fghrtypl^2 \\\\\n&= 2fghrtypl ujmnbhyt + iklopplm ujmnbhyt + fghrtypl^2 \\\\\n&= ujmnbhyt^2 + fghrtypl^2 = hygtfrde,\n\\end{align*}\nand similarly $2asdfghjk+cvbnmklj = zxcvbnml$, so that $\\{ghjklpoi\\}$ satisfies\nthe same recurrence as $\\{fghrtypl\\}$.  Since further $edcrfvgb=1, wsxderfc=2$ (where\nwe use the recurrence for $\\{fghrtypl\\}$ to calculate $poiuytre=0$),\nwe deduce that $ghjklpoi=fghrtypl$ for all $plmoknji$.  In particular,\n$fghrtypl^2+ujmnbhyt^2 = hygtfrde = zxcasdqw$.\n\nSecond solution:\nNote that\n\\begin{multline*}\n\\frac{1}{1-2ksjdhfla-ksjdhfla^2} \\\\\n\\qquad = \\frac{1}{2\\sqrt{2}} \\left(\n\\frac{\\sqrt{2}+1}{1-(1+\\sqrt{2})ksjdhfla} + \\frac{\\sqrt{2}-1}{1-(1-\\sqrt{2})ksjdhfla}\n\\right)\n\\end{multline*}\nand that\n\\[\n\\frac{1}{1 + (1\\pm\\sqrt{2})ksjdhfla} = \\sum_{plmoknji=0}^\\infty (1\\pm\\sqrt{2})^{plmoknji} ksjdhfla^{plmoknji},\n\\]\nso that\n\\[\nfghrtypl = \\frac{1}{2\\sqrt{2}} \\left((\\sqrt{2}+1)^{plmoknji+1} - (1-\\sqrt{2})^{plmoknji+1}\n\\right).\n\\]\nA simple computation (omitted here) now shows that\n$fghrtypl^2 + ujmnbhyt^2 = zxcasdqw$.\n\nThird solution (by Richard Stanley):\nLet $plkjhytg$ be the matrix $\\begin{pmatrix} 0 & 1 \\\\1 & 2 \\end{pmatrix}$.\nA simple induction argument shows that\n\\[\nplkjhytg^{plmoknji+2} = \\begin{pmatrix} fghrtypl & ujmnbhyt \\\\ ujmnbhyt & tyuiohgf\n\\end{pmatrix}.\n\\]\nThe desired result now follows from comparing the top left corner\nentries of the equality $plkjhytg^{plmoknji+2} plkjhytg^{plmoknji+2} = plkjhytg^{2plmoknji+4}$. "
    },
    "kernel_variant": {
      "question": "Let $s\\in\\mathbf Z\\setminus\\{0\\}$ and let $\\bigl(a_n\\bigr)_{n\\ge 0}$ be the \\emph{unique} integer sequence given by  \n\\[\na_{0}=1,\\qquad a_{1}=s,\\qquad   \na_{n+1}=s\\,a_{n}+a_{n-1}\\quad(n\\ge 1),\n\\qquad  \n\\Bigl(\\;\\sum_{n\\ge 0}a_nx^{\\,n}=\\dfrac{1}{1-sx-x^{2}}\\Bigr).\n\\]\n\n1. (Existence) Show that for every $n\\ge 0$ the quadratic form $a_n^{\\,2}+a_{n+1}^{\\,2}$ reappears in the sequence, i.e.  \n\\[\n\\exists\\,m\\ge 0:\\; a_n^{\\,2}+a_{n+1}^{\\,2}=a_m .\n\\]\n\n2. (Uniqueness) Prove that the index $m$ from part 1 is unique and equals  \n\\[\nm=2n+2\\qquad(n\\ge 0).\n\\]\n\n3. (Rigidity) Conversely, let $\\bigl(b_n\\bigr)_{n\\ge 0}$ be a \\emph{non-zero} integer sequence which  \n\n(i) satisfies a homogeneous linear recurrence of order two with non-vanishing integer coefficients  \n\\[\nb_{\\,n+1}=p\\,b_{\\,n}+q\\,b_{\\,n-1}\\quad(n\\ge 1),\\qquad q\\neq 0,\\;p\\neq 0,\n\\]\n\n(ii) fulfils the quadratic identity  \n\\[\nb_n^{\\,2}+b_{n+1}^{\\,2}=b_{\\,2n+2}\\qquad\\forall\\,n\\ge 0 .\n\\]\n\nShow that necessarily $q=1$, $p=s$ for a \\emph{unique} $s\\in\\mathbf Z\\setminus\\{0\\}$ and that the whole sequence coincides with $\\bigl(a_n\\bigr)$; equivalently  \n\\[\n\\sum_{n\\ge 0} b_nx^{\\,n}= \\dfrac{1}{1-sx-x^{2}}\\,. \n\\]\n\n\\bigskip",
      "solution": "\\textbf{0.\\;A $2\\times 2$ matrix model.}  \nFor $s\\in\\mathbf Z\\setminus\\{0\\}$ put  \n\\[\nA_s=\\begin{pmatrix}0&1\\\\ 1&s\\end{pmatrix},\\qquad\\det A_s=-1 .\n\\]\nInduction on $n$ gives  \n\\[\nA_s^{\\,n+2}=\\begin{pmatrix}a_n&a_{n+1}\\\\[2pt] a_{n+1}&a_{n+2}\\end{pmatrix}\\qquad(n\\ge 0).\n\\tag{1}\n\\]\n\n\\textbf{1.\\;Existence of the index.}  \nSquaring \\eqref{1} and using $A_s^{\\,n+2}A_s^{\\,n+2}=A_s^{\\,2n+4}$ yields\n\\[\na_n^{\\,2}+a_{n+1}^{\\,2}=a_{\\,2n+2}\\qquad(n\\ge 0),\n\\]\nso $m=2n+2$ always works.\n\n\\textbf{2.\\;Uniqueness of the index.}\n\n\\emph{2.1  A uniform Binet formula.}  \nLet $\\alpha,\\beta$ be the roots of $t^{2}-s\\,t-1=0$.  Then  \n\\[\n\\alpha\\beta=-1,\\qquad |\\alpha|>1>|\\beta|,\\qquad \\beta=-\\alpha^{-1},\n\\]\nand  \n\\[\na_n=\\frac{\\alpha^{\\,n+1}-\\beta^{\\,n+1}}{\\alpha-\\beta}\\qquad(n\\ge 0).\n\\tag{2}\n\\]\n\n\\emph{2.2  Comparison of dominant terms.}  \nAssume $a_n^{\\,2}+a_{n+1}^{\\,2}=a_m$.  Insert \\eqref{2} and clear denominators to obtain  \n\\[\n\\alpha^{\\,2n+4}+\\alpha^{\\,2n+2}+\\alpha^{-2n-2}+\\alpha^{-2n-4}\n     =(\\alpha+\\alpha^{-1})\\bigl(\\alpha^{\\,m+1}-(-1)^{\\,m+1}\\alpha^{-m-1}\\bigr).\n\\tag{3}\n\\]\n\nBecause $|\\alpha|>1$, the monomials on the two sides of \\eqref{3} with maximal absolute value are   \n$\\alpha^{\\,2n+4}$ on the left and $\\alpha^{\\,m+2}$ on the right. \nThe coefficient of $\\alpha^{\\,2n+4}$ on the left equals $1$, while the coefficient of $\\alpha^{\\,m+2}$ on the right is $\\alpha+\\alpha^{-1}\\neq 0$.  \nHence equality of the two Laurent polynomials forces the exponents to coincide:\n\\[\n2n+4=m+2\\quad\\Longrightarrow\\quad m=2n+2.\n\\]\nTherefore the index is unique.\n\n\\textbf{3.\\;Classification of all integer sequences satisfying (i)–(ii) with $p\\neq 0$.}\n\nLet $\\bigl(b_n\\bigr)$ be a non-zero sequence fulfilling (i) and (ii).\n\n\\emph{3.1  Binet representation.}  \nLet $r_1,r_2$ be the roots of $t^{2}-p\\,t-q=0$:\n\\[\nr_1+r_2=p\\neq 0,\\qquad r_1r_2=-q\\neq 0.\n\\tag{4}\n\\]\nBecause $q\\neq 0$,  \n\\[\nb_n=\\begin{cases}\nA\\,r_1^{\\,n}+B\\,r_2^{\\,n},&\\text{if }r_1\\neq r_2,\\\\[4pt]\n(C+Dn)\\,r^{\\,n},&\\text{if }r_1=r_2=r,\n\\end{cases}\\qquad n\\ge 0,\n\\tag{5}\n\\]\nwith (complex) constants $A,B$ (resp. $C,D$) not both zero.\n\n\\smallskip\n\\emph{3.2  The parameter $q$ equals $1$.}\n\n\\underline{Case $r_1\\neq r_2$.}\n\n(1) \\emph{Both coefficients $A,B$ are non-zero.}  \nIf, say, $A=0$, then $b_n=B\\,r_2^{\\,n}$, and (ii) at $n=0$ gives  \n$B^{\\,2}=B\\,r_2^{\\,2}$, whence $B=0$—contradiction.  Thus $AB\\neq 0$.\n\n(2) \\emph{Plugging \\eqref{5} into (ii).}  \nUsing $r_1r_2=-q$ we find  \n\\[\n\\bigl[A^{\\,2}(1+r_1^{\\,2})-A\\,r_1^{\\,2}\\bigr]\\,r_1^{\\,2n}\n+\n\\bigl[B^{\\,2}(1+r_2^{\\,2})-B\\,r_2^{\\,2}\\bigr]\\,r_2^{\\,2n}\n+\n2AB(1-q)(-q)^{\\,n}=0\\qquad(n\\ge 0).\n\\tag{6}\n\\]\n\n(3) \\emph{Linear independence of the three exponential subsequences.}  \nBecause $p\\neq 0$, we have $r_2\\neq -r_1$, hence $u=r_1^{\\,2}$ and $v=r_2^{\\,2}$ are distinct.  \nMoreover $w=-q$ is different from $u$ and $v$ (otherwise $r_1=r_2$).  \nTherefore the sequences $u^{\\,n}$, $v^{\\,n}$ and $w^{\\,n}$ are linearly independent over $\\mathbf C$; equality \\eqref{6} forces the three coefficients to vanish:\n\\[\nA^{\\,2}(1+r_1^{\\,2})-A\\,r_1^{\\,2}=0,\\quad\nB^{\\,2}(1+r_2^{\\,2})-B\\,r_2^{\\,2}=0,\\quad\n1-q=0.\n\\]\nThus $q=1$.\n\n\\underline{Case $r_1=r_2$ (repeated root).}  \n\nHere $p^{\\,2}+4q=0$, so $q<0$ and $p\\neq 0$.  \nWrite $b_n=(C+Dn)\\,r^{\\,n}$ as in \\eqref{5}.  Substituting into (ii) and comparing the coefficients of $n^{\\,2}$ gives $D=0$.  \nHence $b_n=C\\,r^{\\,n}$ is geometric; (ii) with $n=0$ then forces $C=0$, contradicting non-triviality.  Therefore the repeated-root case is impossible.\n\nConsequently $q=1$ in all cases.\n\n\\medskip\n\\emph{3.3  Determination of $p$ and the initial data.}  \nWith $q=1$, relation (ii) at $n=0$ reads\n\\[\nb_0^{\\,2}+b_1^{\\,2}=b_2=p\\,b_1+b_0.\n\\tag{7}\n\\]\n\n\\underline{Step 1: $b_0\\neq 0$.}  \nIf $b_0=0$, then \\eqref{7} gives $b_1^{\\,2}=p\\,b_1$, hence $b_1=p\\neq 0$.  The recurrence yields\n$b_2=p^{\\,2}$, $b_3=p^{\\,3}+p$, and (ii) at $n=1$ fails as before; thus $b_0\\neq 0$.\n\n\\underline{Step 2: a \\emph{pair} of identities.}  \nBesides \\eqref{7} we need the identity with $n=1$:\n\\[\nb_1^{\\,2}+b_2^{\\,2}=b_4.\n\\tag{8}\n\\]\nUsing the recurrence $b_2=p\\,b_1+b_0$, $b_3=p\\,b_2+b_1$, $b_4=p\\,b_3+b_2$ we obtain from \\eqref{8} after a short calculation\n\\[\np\\,b_1(2b_0-1)=p^{\\,2}b_0^{\\,2}.\n\\tag{9}\n\\]\n\n\\underline{Step 3: elimination of $b_0\\neq 1$.}  \nIf $b_0\\neq 1$, then $2b_0-1$ is coprime to $b_0$; from \\eqref{9} it therefore divides $p$.  \nSolving \\eqref{9} for $p$ and inserting the result into \\eqref{7} gives\n\\[\n(b_0-1)\\bigl[b_0^{\\,3}+b_1^{\\,2}(b_0-1)\\bigr]=0.\n\\]\nBecause $b_0\\neq 1$, the second factor must vanish, which is impossible:\n\n• If $b_0>1$, both summands are positive.  \n• If $b_0<0$, both summands are negative (and $b_1=0$ would make the\nsequence trivial, excluded).\n\nHence $b_0=1$ is forced.\n\n\\underline{Step 4: conclusion.}  \nWith $b_0=1$, equation \\eqref{7} becomes\n\\[\n1+b_1^{\\,2}=p\\,b_1+1\\quad\\Longrightarrow\\quad b_1^{\\,2}=p\\,b_1.\n\\]\nBecause $p\\neq 0$ the only solution is $b_1=p=:s\\in\\mathbf Z\\setminus\\{0\\}$.\n\n\\medskip\n\\emph{3.4  Identification with $\\bigl(a_n\\bigr)$.}  \nWith $(b_0,b_1)=(1,s)$ and parameters $(p,q)=(s,1)$ the recurrence  \n\\[\nb_{\\,n+1}=s\\,b_{\\,n}+b_{\\,n-1}\\quad(n\\ge 1)\n\\]\ncoincides with the defining relation of $\\bigl(a_n\\bigr)$.  \nInduction therefore gives\n\\[\nb_n=a_n\\qquad\\forall\\,n\\ge 0,\n\\]\nand the generating series of $\\bigl(b_n\\bigr)$ equals $(1-sx-x^{2})^{-1}$ as claimed.\n\n\\hfill$\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.764976",
        "was_fixed": false,
        "difficulty_analysis": "• The original and kernel problems asked only to prove the existence of some index $m$ (and to determine it) for one particular quadratic sequence; here we must establish  \n   (a) the identity for every integer parameter $s$ with $|s|\\ge 2$,  \n   (b) prove that *no other* value of $m$ can work, and  \n   (c) classify *all* second-order recurrences for which the identity $u_n^{2}+u_{n+1}^{2}=u_{2n+2}$ holds.  \n\n• Part 3 forces the solver to reverse-engineer the recurrence from the quadratic\n   identity, bringing in asymptotic analysis of Binet forms, linear-algebraic\n   arguments with companion matrices, and an invariant–bilinear-form\n   computation.  None of these tools is needed in the original problem.\n\n• The solution therefore mixes several advanced techniques—matrix\n   exponentiation, eigenvalue asymptotics, rigidity of bilinear\n   invariants—turning a one-line identity into a full characterisation\n   theorem.  This combination of analytic, algebraic and structural arguments\n   adds several non-trivial layers of complexity beyond both the original\n   problem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $s\\in\\mathbf{Z}\\setminus\\{0\\}$ and consider the \\emph{unique} integer sequence $\\bigl(a_n\\bigr)_{n\\ge 0}$ defined by  \n\\[\na_0=1,\\qquad a_1=s,\\qquad \na_{n+1}=s\\,a_{n}+a_{\\,n-1}\\quad(n\\ge 1),\n\\qquad \n\\Bigl(\\;\\sum_{n\\ge 0}a_nx^{\\,n}=\\frac{1}{1-sx-x^{2}}\\Bigr).\n\\]\n\n1.\\;(\\,Existence\\,) Show that for every $n\\ge 0$ the quadratic form $a_n^{\\,2}+a_{n+1}^{\\,2}$ itself occurs in the sequence, that is\n\\[\n\\exists\\,m\\ge 0\\colon\\;a_n^{\\,2}+a_{n+1}^{\\,2}=a_m .\n\\]\n\n2.\\;(\\,Uniqueness\\,) Prove that this index is unique and equals  \n\\[\nm=2n+2\\qquad(n\\ge 0).\n\\]\n\n3.\\;(\\,Rigidity\\,) Conversely, let $\\bigl(b_n\\bigr)_{n\\ge 0}$ be a \\emph{non–zero} integer sequence which  \n\n(i) satisfies a homogeneous linear recurrence of order two with constant integer coefficients  \n\\[\nb_{\\,n+1}=p\\,b_{\\,n}+q\\,b_{\\,n-1}\\quad(n\\ge 1),\\qquad q\\neq 0,\\;p\\neq 0,\n\\]\n\n(ii) fulfils the quadratic identity  \n\\[\nb_n^{\\,2}+b_{n+1}^{\\,2}=b_{\\,2n+2}\\qquad\\forall\\,n\\ge 0 .\n\\]\n\nShow that necessarily $q=1$, $p=s$ for a \\emph{unique} $s\\in\\mathbf{Z}\\setminus\\{0\\}$ and that the whole sequence coincides with $\\bigl(a_n\\bigr)$; equivalently  \n\\[\n\\sum_{n\\ge 0} b_nx^{\\,n}= \\frac{1}{1-sx-x^{2}}\\,. \n\\]\n\n\\bigskip",
      "solution": "\\textbf{0.\\;A $2\\times 2$ matrix model.}  \nFor $s\\in\\mathbf{Z}\\setminus\\{0\\}$ put  \n\\[\nA_s=\\begin{pmatrix}0&1\\\\ 1&s\\end{pmatrix},\\qquad\\det A_s=-1 .\n\\]\nA short induction gives  \n\\[\nA_s^{\\,n+2}=\\begin{pmatrix}a_n&a_{n+1}\\\\[2pt] a_{n+1}&a_{n+2}\\end{pmatrix}\\qquad(n\\ge 0).\n\\tag{1}\n\\]\n\n\\textbf{1.\\;Existence of the index.}  \nSquaring \\eqref{1} and using $A_s^{\\,n+2}A_s^{\\,n+2}=A_s^{\\,2n+4}$ yields\n\\[\na_n^{\\,2}+a_{n+1}^{\\,2}=a_{\\,2n+2}\\qquad(n\\ge 0),\n\\]\nso $m=2n+2$ always works.\n\n\\textbf{2.\\;Uniqueness of the index.}\n\n\\emph{2.1  A uniform Binet formula.}  \nLet $\\alpha,\\beta$ be the roots of $t^{2}-s\\,t-1=0$.  Then  \n\\[\n\\alpha\\beta=-1,\\qquad |\\alpha|>1>|\\beta|,\\qquad \\beta=-\\alpha^{-1},\n\\]\nand  \n\\[\na_n=\\frac{\\alpha^{\\,n+1}-\\beta^{\\,n+1}}{\\alpha-\\beta}\\qquad(n\\ge 0).\n\\tag{2}\n\\]\n\n\\emph{2.2  Comparison of dominant terms.}  \nAssume $a_n^{\\,2}+a_{n+1}^{\\,2}=a_m$.  Inserting \\eqref{2} and clearing denominators one obtains the identity  \n\\[\n\\alpha^{\\,2n+4}+\\alpha^{\\,2n+2}+\\alpha^{-2n-2}+\\alpha^{-2n-4}\n     =(\\alpha+\\alpha^{-1})\\bigl(\\alpha^{\\,m+1}-(-1)^{\\,m+1}\\alpha^{-m-1}\\bigr).\n\\]\nBecause $|\\alpha|>1$, the monomial of maximal absolute value on the left is $\\alpha^{\\,2n+4}$ whilst on the right it is $\\alpha^{\\,m+2}$.  Equality therefore forces  \n\\[\n2n+4=m+2\\quad\\Longrightarrow\\quad m=2n+2,\n\\]\nwhence the index is unique.\n\n\\textbf{3.\\;Classification of all integer sequences satisfying (i)–(ii) under the non–degeneracy condition $p\\neq 0$.}\n\n\\medskip\nLet $\\bigl(b_n\\bigr)$ be a non–zero sequence fulfilling (i) and (ii).\n\n\\emph{3.1  Binet representation.}  \nLet $r_1,r_2$ be the roots of $t^{2}-p\\,t-q=0$:\n\\[\nr_1+r_2=p\\neq 0,\\qquad r_1r_2=-q\\neq 0.\n\\tag{3}\n\\]\nBecause $q\\neq 0$,  \n\\[\nb_n=\\begin{cases}\nA\\,r_1^{\\,n}+B\\,r_2^{\\,n},&\\text{if }r_1\\neq r_2,\\\\[4pt]\n(C+Dn)\\,r^{\\,n},&\\text{if }r_1=r_2=r,\n\\end{cases}\\qquad n\\ge 0,\n\\tag{4}\n\\]\nwith \\emph{complex} constants $A,B$ (resp. $C,D$) not both zero.\n\n\\medskip\n\\emph{3.2  The parameter $q$ equals $1$.}\n\n\\underline{Case $r_1\\neq r_2$.}\n\n\\smallskip\n(1) \\emph{Both coefficients $A,B$ are non–zero.}  \nIf, for instance, $A=0$, then $b_n=B\\,r_2^{\\,n}$, and (ii) at $n=0$ gives  \n$B^{\\,2}=B\\,r_2^{\\,2}$, whence $B=0$—contradiction.  Thus $AB\\neq 0$.\n\n\\smallskip\n(2) \\emph{Plugging \\eqref{4} into (ii).}  \nUsing $r_1r_2=-q$ we arrive at  \n\\[\n\\bigl[A^{\\,2}(1+r_1^{\\,2})-A\\,r_1^{\\,2}\\bigr]\\,r_1^{\\,2n}\n+\n\\bigl[B^{\\,2}(1+r_2^{\\,2})-B\\,r_2^{\\,2}\\bigr]\\,r_2^{\\,2n}\n+\n2AB(1-q)(-q)^{\\,n}=0\\qquad(n\\ge 0).\n\\tag{5}\n\\]\n\n\\smallskip\n(3) \\emph{Linear independence of the three exponential subsequences.}  \nBecause $p\\neq 0$, we have $r_2\\neq -r_1$, so $u=r_1^{\\,2}$ and $v=r_2^{\\,2}$ are \\emph{distinct}.  \nMoreover $w=-q$ cannot coincide with either $u$ or $v$; indeed, $u=w$ would imply $r_1^{\\,2}=-q=r_1r_2$, hence $r_1=r_2$ contrary to the present case, and similarly for $v=w$.  \nThus the sequences $u^{\\,n},v^{\\,n},w^{\\,n}$ are linearly independent over $\\mathbf{C}$; equality \\eqref{5} forces the three coefficients to vanish:\n\\[\nA^{\\,2}(1+r_1^{\\,2})-A\\,r_1^{\\,2}=0,\\quad\nB^{\\,2}(1+r_2^{\\,2})-B\\,r_2^{\\,2}=0,\\quad\n1-q=0.\n\\]\nConsequently $q=1$.\n\n\\underline{Case $r_1=r_2$ (repeated root).}  \n\nHere $p^{\\,2}+4q=0$, hence $q<0$ and $p\\neq 0$.  \nWrite $b_n=(C+Dn)\\,r^{\\,n}$ as in \\eqref{4}.  Substituting into (ii) and comparing the coefficients of $n^{\\,2}$ gives $D=0$.  \nThus $b_n=C\\,r^{\\,n}$ is geometric; (ii) with $n=0$ then forces $C=0$, contradicting non–triviality.  The repeated–root case is impossible.\n\n\\medskip\nTherefore $q=1$ in all cases.\n\n\\medskip\n\\emph{3.3  Determination of $p$ and the initial data.}  \nWith $q=1$, relation (ii) at $n=0$ reads\n\\[\nb_0^{\\,2}+b_1^{\\,2}=b_2=p\\,b_1+b_0.\n\\tag{6}\n\\]\nBecause $p\\neq 0$, $b_1$ cannot be $0$: if $b_1=0$, then \\eqref{6} yields $b_0^{\\,2}=b_0$, hence $b_0=1$, and the identity at $n=1$ would give $b_2^{\\,2}=b_4$; a quick calculation using the recurrence shows this forces $p=0$, contradicting the hypothesis.  \nConsequently $b_1\\neq 0$ and \\eqref{6} gives $p=b_1=:s\\in\\mathbf{Z}\\setminus\\{0\\}$ as well as $b_0=1$.  \n\n\\medskip\n\\emph{3.4  Identification with $\\bigl(a_n\\bigr)$.}  \nWith $(b_0,b_1)=(1,s)$ and parameters $(p,q)=(s,1)$, the recurrence  \n\\[\nb_{\\,n+1}=s\\,b_{\\,n}+b_{\\,n-1}\\quad(n\\ge 1)\n\\]\ncoincides with the defining relation of $\\bigl(a_n\\bigr)$.  \nInduction therefore gives\n\\[\nb_n=a_n\\qquad\\forall\\,n\\ge 0,\n\\]\nand the generating series of $\\bigl(b_n\\bigr)$ equals $(1-sx-x^{2})^{-1}$ as claimed.\n\n\\hfill$\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.586538",
        "was_fixed": false,
        "difficulty_analysis": "• The original and kernel problems asked only to prove the existence of some index $m$ (and to determine it) for one particular quadratic sequence; here we must establish  \n   (a) the identity for every integer parameter $s$ with $|s|\\ge 2$,  \n   (b) prove that *no other* value of $m$ can work, and  \n   (c) classify *all* second-order recurrences for which the identity $u_n^{2}+u_{n+1}^{2}=u_{2n+2}$ holds.  \n\n• Part 3 forces the solver to reverse-engineer the recurrence from the quadratic\n   identity, bringing in asymptotic analysis of Binet forms, linear-algebraic\n   arguments with companion matrices, and an invariant–bilinear-form\n   computation.  None of these tools is needed in the original problem.\n\n• The solution therefore mixes several advanced techniques—matrix\n   exponentiation, eigenvalue asymptotics, rigidity of bilinear\n   invariants—turning a one-line identity into a full characterisation\n   theorem.  This combination of analytic, algebraic and structural arguments\n   adds several non-trivial layers of complexity beyond both the original\n   problem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}