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{ | |
"cells": [ | |
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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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