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    "### Property of Beatty sequence and its complementary:\n",
    "Every positive integer belongs to exactly one of ${\\mathcal {B}}_{r}^{(i)}$ or ${\\mathcal {B}}_{s}^{(i)}$.\n",
    "\n",
    "\n",
    "    \n",
    "$S(n,r) + S(\\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor,s) = \\sum_{i=1}^{{\\mathcal {B}}_{r}^{(n)}} i = \\frac{{\\mathcal {B}}_{r}^{(n)}({\\mathcal {B}}_{r}^{(n)}+1)}{2} \\qquad \\textbf{(ii)}$\n",
    "\n",
    "$\\stackrel{using  \\; (i)}{=>} S(n,r) + S(\\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor,r) + \\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor(\\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor+1)$\n",
    "\n",
    "$\\stackrel{using  \\; (ii)}{=>} S(n,r) = \\frac{{\\mathcal {B}}_{r}^{(n)}({\\mathcal {B}}_{r}^{(n)}+1)}{2} - S(\\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor,r) - \\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor(\\lfloor{\\frac{{\\mathcal {B}}_{r}^{(n)}}{s}}\\rfloor+1)$"
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