பல்கோண எண்: திருத்தங்களுக்கு இடையிலான வேறுபாடு
No edit summary |
No edit summary |
||
வரிசை 222: | வரிசை 222: | ||
<table border="3" cellpadding="3"><tr><b> |
<table border="3" cellpadding="3"><tr><b> |
||
<td align=center>பெயர்</td><td align=center>Formula</td><td align=center><i>n</i>=1</td><td align=center>2</td><td align=center>3</td><td align=center>4</td><td align=center>5</td><td align=center>6</td><td align=center>7</td><td align=center>8</td><td align=center>9</td><td align=center>10</td><td align=center>11</td><td align=center>12</td><td align=center>13</td></b></tr> |
<td align=center>பெயர்</td><td align=center>Formula</td><td align=center><i>n</i>=1</td><td align=center>2</td><td align=center>3</td><td align=center>4</td><td align=center>5</td><td align=center>6</td><td align=center>7</td><td align=center>8</td><td align=center>9</td><td align=center>10</td><td align=center>11</td><td align=center>12</td><td align=center>13</td></b></tr> |
||
<tr><td align=center> |
<tr><td align=center>முக்கோணம்</td><td align=center>½<i>n</i>(1<i>n</i> + 1)</td> |
||
<td align=right>1</td><td align=right>3</td><td align=right>6</td><td align=right>10</td><td align=right>15</td><td align=right>21</td><td align=right>28</td><td align=right>36</td><td align=right>45</td><td align=right>55</td><td align=right>66</td><td align=right>78</td><td align=right>91</td></tr> |
<td align=right>1</td><td align=right>3</td><td align=right>6</td><td align=right>10</td><td align=right>15</td><td align=right>21</td><td align=right>28</td><td align=right>36</td><td align=right>45</td><td align=right>55</td><td align=right>66</td><td align=right>78</td><td align=right>91</td></tr> |
||
<tr><td align=center>சதுரம்</td><td align=center>½<i>n</i>(2<i>n</i> - 0)</td> |
<tr><td align=center>சதுரம்</td><td align=center>½<i>n</i>(2<i>n</i> - 0)</td> |
||
<td align=right>1</td><td align=right>4</td><td align=right>9</td><td align=right>16</td><td align=right>25</td><td align=right>36</td><td align=right>49</td><td align=right>64</td><td align=right>81</td><td align=right>100</td><td align=right>121</td><td align=right>144</td><td align=right>169</td></tr> |
<td align=right>1</td><td align=right>4</td><td align=right>9</td><td align=right>16</td><td align=right>25</td><td align=right>36</td><td align=right>49</td><td align=right>64</td><td align=right>81</td><td align=right>100</td><td align=right>121</td><td align=right>144</td><td align=right>169</td></tr> |
||
<tr><td align=center> |
<tr><td align=center>ஐங்கோணம்</td><td align=center>½<i>n</i>(3<i>n</i> - 1)</td> |
||
<td align=right>1</td><td align=right>5</td><td align=right>12</td><td align=right>22</td><td align=right>35</td><td align=right>51</td><td align=right>70</td><td align=right>92</td><td align=right>117</td><td align=right>145</td><td align=right>176</td><td align=right>210</td><td align=right>247</td></tr> |
<td align=right>1</td><td align=right>5</td><td align=right>12</td><td align=right>22</td><td align=right>35</td><td align=right>51</td><td align=right>70</td><td align=right>92</td><td align=right>117</td><td align=right>145</td><td align=right>176</td><td align=right>210</td><td align=right>247</td></tr> |
||
<tr><td align=center>அறுகோணம்</td><td align=center>½<i>n</i>(4<i>n</i> - 2)</td> |
<tr><td align=center>அறுகோணம்</td><td align=center>½<i>n</i>(4<i>n</i> - 2)</td> |
||
<td align=right>1</td><td align=right>6</td><td align=right>15</td><td align=right>28</td><td align=right>45</td><td align=right>66</td><td align=right>91</td><td align=right>120</td><td align=right>153</td><td align=right>190</td><td align=right>231</td><td align=right>276</td><td align=right>325</td></tr> |
<td align=right>1</td><td align=right>6</td><td align=right>15</td><td align=right>28</td><td align=right>45</td><td align=right>66</td><td align=right>91</td><td align=right>120</td><td align=right>153</td><td align=right>190</td><td align=right>231</td><td align=right>276</td><td align=right>325</td></tr> |
||
<tr><td align=center> |
<tr><td align=center>எழுகோணம்</td><td align=center>½<i>n</i>(5<i>n</i> - 3)</td> |
||
<td align=right>1</td><td align=right>7</td><td align=right>18</td><td align=right>34</td><td align=right>55</td><td align=right>81</td><td align=right>112</td><td align=right>148</td><td align=right>189</td><td align=right>235</td><td align=right>286</td><td align=right>342</td><td align=right>403</td></tr> |
<td align=right>1</td><td align=right>7</td><td align=right>18</td><td align=right>34</td><td align=right>55</td><td align=right>81</td><td align=right>112</td><td align=right>148</td><td align=right>189</td><td align=right>235</td><td align=right>286</td><td align=right>342</td><td align=right>403</td></tr> |
||
<tr><td align=center> |
<tr><td align=center>எண்கோணம்</td><td align=center>½<i>n</i>(6<i>n</i> - 4)</td> |
||
<td align=right>1</td><td align=right>8</td><td align=right>21</td><td align=right>40</td><td align=right>65</td><td align=right>96</td><td align=right>133</td><td align=right>176</td><td align=right>225</td><td align=right>280</td><td align=right>341</td><td align=right>408</td><td align=right>481</td></tr> |
<td align=right>1</td><td align=right>8</td><td align=right>21</td><td align=right>40</td><td align=right>65</td><td align=right>96</td><td align=right>133</td><td align=right>176</td><td align=right>225</td><td align=right>280</td><td align=right>341</td><td align=right>408</td><td align=right>481</td></tr> |
||
<tr><td align=center> |
<tr><td align=center>நவகோணம்</td><td align=center>½<i>n</i>(7<i>n</i> - 5)</td> |
||
<td align=right>1</td><td align=right>9</td><td align=right>24</td><td align=right>46</td><td align=right>75</td><td align=right>111</td><td align=right>154</td><td align=right>204</td><td align=right>261</td><td align=right>325</td><td align=right>396</td><td align=right>474</td><td align=right>559</td></tr> |
<td align=right>1</td><td align=right>9</td><td align=right>24</td><td align=right>46</td><td align=right>75</td><td align=right>111</td><td align=right>154</td><td align=right>204</td><td align=right>261</td><td align=right>325</td><td align=right>396</td><td align=right>474</td><td align=right>559</td></tr> |
||
<tr><td align=center> |
<tr><td align=center>தசகோணம்</td><td align=center>½<i>n</i>(8<i>n</i> - 6)</td> |
||
<td align=right>1</td><td align=right>10</td><td align=right>27</td><td align=right>52</td><td align=right>85</td><td align=right>126</td><td align=right>175</td><td align=right>232</td><td align=right>297</td><td align=right>370</td><td align=right>451</td><td align=right>540</td><td align=right>637</td></tr> |
<td align=right>1</td><td align=right>10</td><td align=right>27</td><td align=right>52</td><td align=right>85</td><td align=right>126</td><td align=right>175</td><td align=right>232</td><td align=right>297</td><td align=right>370</td><td align=right>451</td><td align=right>540</td><td align=right>637</td></tr> |
||
<tr><td align=center>11- |
<tr><td align=center>11-கோணம்</td><td align=center>½<i>n</i>(9<i>n</i> - 7)</td> |
||
<td align=right>1</td><td align=right>11</td><td align=right>30</td><td align=right>58</td><td align=right>95</td><td align=right>141</td><td align=right>196</td><td align=right>260</td><td align=right>333</td><td align=right>415</td><td align=right>506</td><td align=right>606</td><td align=right>715</td></tr> |
<td align=right>1</td><td align=right>11</td><td align=right>30</td><td align=right>58</td><td align=right>95</td><td align=right>141</td><td align=right>196</td><td align=right>260</td><td align=right>333</td><td align=right>415</td><td align=right>506</td><td align=right>606</td><td align=right>715</td></tr> |
||
<tr><td align=center>12- |
<tr><td align=center>12-கோணம்</td><td align=center>½<i>n</i>(10<i>n</i> - 8)</td> |
||
<td align=right>1</td><td align=right>12</td><td align=right>33</td><td align=right>64</td><td align=right>105</td><td align=right>156</td><td align=right>217</td><td align=right>288</td><td align=right>369</td><td align=right>460</td><td align=right>561</td><td align=right>672</td><td align=right>793</td></tr> |
<td align=right>1</td><td align=right>12</td><td align=right>33</td><td align=right>64</td><td align=right>105</td><td align=right>156</td><td align=right>217</td><td align=right>288</td><td align=right>369</td><td align=right>460</td><td align=right>561</td><td align=right>672</td><td align=right>793</td></tr> |
||
<tr><td align=center>13- |
<tr><td align=center>13-கோணம்</td><td align=center>½<i>n</i>(11<i>n</i> - 9)</td> |
||
<td align=right>1</td><td align=right>13</td><td align=right>36</td><td align=right>70</td><td align=right>115</td><td align=right>171</td><td align=right>238</td><td align=right>316</td><td align=right>405</td><td align=right>505</td><td align=right>616</td><td align=right>738</td><td align=right>871</td></tr> |
<td align=right>1</td><td align=right>13</td><td align=right>36</td><td align=right>70</td><td align=right>115</td><td align=right>171</td><td align=right>238</td><td align=right>316</td><td align=right>405</td><td align=right>505</td><td align=right>616</td><td align=right>738</td><td align=right>871</td></tr> |
||
<tr><td align=center>14- |
<tr><td align=center>14-கோணம்</td><td align=center>½<i>n</i>(12<i>n</i> - 10)</td> |
||
<td align=right>1</td><td align=right>14</td><td align=right>39</td><td align=right>76</td><td align=right>125</td><td align=right>186</td><td align=right>259</td><td align=right>344</td><td align=right>441</td><td align=right>550</td><td align=right>671</td><td align=right>804</td><td align=right>949</td></tr> |
<td align=right>1</td><td align=right>14</td><td align=right>39</td><td align=right>76</td><td align=right>125</td><td align=right>186</td><td align=right>259</td><td align=right>344</td><td align=right>441</td><td align=right>550</td><td align=right>671</td><td align=right>804</td><td align=right>949</td></tr> |
||
<tr><td align=center>15- |
<tr><td align=center>15-கோணம்</td><td align=center>½<i>n</i>(13<i>n</i> - 11)</td> |
||
<td align=right>1</td><td align=right>15</td><td align=right>42</td><td align=right>82</td><td align=right>135</td><td align=right>201</td><td align=right>280</td><td align=right>372</td><td align=right>477</td><td align=right>595</td><td align=right>726</td><td align=right>870</td><td align=right>1027</td></tr> |
<td align=right>1</td><td align=right>15</td><td align=right>42</td><td align=right>82</td><td align=right>135</td><td align=right>201</td><td align=right>280</td><td align=right>372</td><td align=right>477</td><td align=right>595</td><td align=right>726</td><td align=right>870</td><td align=right>1027</td></tr> |
||
<tr><td align=center>16- |
<tr><td align=center>16-கோணம்</td><td align=center>½<i>n</i>(14<i>n</i> - 12)</td> |
||
<td align=right>1</td><td align=right>16</td><td align=right>45</td><td align=right>88</td><td align=right>145</td><td align=right>216</td><td align=right>301</td><td align=right>400</td><td align=right>513</td><td align=right>640</td><td align=right>781</td><td align=right>936</td><td align=right>1105</td></tr> |
<td align=right>1</td><td align=right>16</td><td align=right>45</td><td align=right>88</td><td align=right>145</td><td align=right>216</td><td align=right>301</td><td align=right>400</td><td align=right>513</td><td align=right>640</td><td align=right>781</td><td align=right>936</td><td align=right>1105</td></tr> |
||
<tr><td align=center>17- |
<tr><td align=center>17-கோணம்</td><td align=center>½<i>n</i>(15<i>n</i> - 13)</td> |
||
<td align=right>1</td><td align=right>17</td><td align=right>48</td><td align=right>94</td><td align=right>155</td><td align=right>231</td><td align=right>322</td><td align=right>428</td><td align=right>549</td><td align=right>685</td><td align=right>836</td><td align=right>1002</td><td align=right>1183</td></tr> |
<td align=right>1</td><td align=right>17</td><td align=right>48</td><td align=right>94</td><td align=right>155</td><td align=right>231</td><td align=right>322</td><td align=right>428</td><td align=right>549</td><td align=right>685</td><td align=right>836</td><td align=right>1002</td><td align=right>1183</td></tr> |
||
<tr><td align=center>18- |
<tr><td align=center>18-கோணம்</td><td align=center>½<i>n</i>(16<i>n</i> - 14)</td> |
||
<td align=right>1</td><td align=right>18</td><td align=right>51</td><td align=right>100</td><td align=right>165</td><td align=right>246</td><td align=right>343</td><td align=right>456</td><td align=right>585</td><td align=right>730</td><td align=right>891</td><td align=right>1068</td><td align=right>1261</td></tr> |
<td align=right>1</td><td align=right>18</td><td align=right>51</td><td align=right>100</td><td align=right>165</td><td align=right>246</td><td align=right>343</td><td align=right>456</td><td align=right>585</td><td align=right>730</td><td align=right>891</td><td align=right>1068</td><td align=right>1261</td></tr> |
||
<tr><td align=center>19- |
<tr><td align=center>19-கோணம்</td><td align=center>½<i>n</i>(17<i>n</i> - 15)</td> |
||
<td align=right>1</td><td align=right>19</td><td align=right>54</td><td align=right>106</td><td align=right>175</td><td align=right>261</td><td align=right>364</td><td align=right>484</td><td align=right>621</td><td align=right>775</td><td align=right>946</td><td align=right>1134</td><td align=right>1339</td></tr> |
<td align=right>1</td><td align=right>19</td><td align=right>54</td><td align=right>106</td><td align=right>175</td><td align=right>261</td><td align=right>364</td><td align=right>484</td><td align=right>621</td><td align=right>775</td><td align=right>946</td><td align=right>1134</td><td align=right>1339</td></tr> |
||
<tr><td align=center>20- |
<tr><td align=center>20-கோணம்</td><td align=center>½<i>n</i>(18<i>n</i> - 16)</td> |
||
<td align=right>1</td><td align=right>20</td><td align=right>57</td><td align=right>112</td><td align=right>185</td><td align=right>276</td><td align=right>385</td><td align=right>512</td><td align=right>657</td><td align=right>820</td><td align=right>1001</td><td align=right>1200</td><td align=right>1417</td></tr> |
<td align=right>1</td><td align=right>20</td><td align=right>57</td><td align=right>112</td><td align=right>185</td><td align=right>276</td><td align=right>385</td><td align=right>512</td><td align=right>657</td><td align=right>820</td><td align=right>1001</td><td align=right>1200</td><td align=right>1417</td></tr> |
||
<tr><td align=center>21- |
<tr><td align=center>21-கோணம்</td><td align=center>½<i>n</i>(19<i>n</i> - 17)</td> |
||
<td align=right>1</td><td align=right>21</td><td align=right>60</td><td align=right>118</td><td align=right>195</td><td align=right>291</td><td align=right>406</td><td align=right>540</td><td align=right>693</td><td align=right>865</td><td align=right>1056</td><td align=right>1266</td><td align=right>1495</td></tr> |
<td align=right>1</td><td align=right>21</td><td align=right>60</td><td align=right>118</td><td align=right>195</td><td align=right>291</td><td align=right>406</td><td align=right>540</td><td align=right>693</td><td align=right>865</td><td align=right>1056</td><td align=right>1266</td><td align=right>1495</td></tr> |
||
<tr><td align=center>22- |
<tr><td align=center>22-கோணம்</td><td align=center>½<i>n</i>(20<i>n</i> - 18)</td> |
||
<td align=right>1</td><td align=right>22</td><td align=right>63</td><td align=right>124</td><td align=right>205</td><td align=right>306</td><td align=right>427</td><td align=right>568</td><td align=right>729</td><td align=right>910</td><td align=right>1111</td><td align=right>1332</td><td align=right>1573</td></tr> |
<td align=right>1</td><td align=right>22</td><td align=right>63</td><td align=right>124</td><td align=right>205</td><td align=right>306</td><td align=right>427</td><td align=right>568</td><td align=right>729</td><td align=right>910</td><td align=right>1111</td><td align=right>1332</td><td align=right>1573</td></tr> |
||
<tr><td align=center>23- |
<tr><td align=center>23-கோணம்</td><td align=center>½<i>n</i>(21<i>n</i> - 19)</td> |
||
<td align=right>1</td><td align=right>23</td><td align=right>66</td><td align=right>130</td><td align=right>215</td><td align=right>321</td><td align=right>448</td><td align=right>596</td><td align=right>765</td><td align=right>955</td><td align=right>1166</td><td align=right>1398</td><td align=right>1651</td></tr> |
<td align=right>1</td><td align=right>23</td><td align=right>66</td><td align=right>130</td><td align=right>215</td><td align=right>321</td><td align=right>448</td><td align=right>596</td><td align=right>765</td><td align=right>955</td><td align=right>1166</td><td align=right>1398</td><td align=right>1651</td></tr> |
||
<tr><td align=center>24- |
<tr><td align=center>24-கோணம்</td><td align=center>½<i>n</i>(22<i>n</i> - 20)</td> |
||
<td align=right>1</td><td align=right>24</td><td align=right>69</td><td align=right>136</td><td align=right>225</td><td align=right>336</td><td align=right>469</td><td align=right>624</td><td align=right>801</td><td align=right>1000</td><td align=right>1221</td><td align=right>1464</td><td align=right>1729</td></tr> |
<td align=right>1</td><td align=right>24</td><td align=right>69</td><td align=right>136</td><td align=right>225</td><td align=right>336</td><td align=right>469</td><td align=right>624</td><td align=right>801</td><td align=right>1000</td><td align=right>1221</td><td align=right>1464</td><td align=right>1729</td></tr> |
||
<tr><td align=center>25- |
<tr><td align=center>25-கோணம்</td><td align=center>½<i>n</i>(23<i>n</i> - 21)</td> |
||
<td align=right>1</td><td align=right>25</td><td align=right>72</td><td align=right>142</td><td align=right>235</td><td align=right>351</td><td align=right>490</td><td align=right>652</td><td align=right>837</td><td align=right>1045</td><td align=right>1276</td><td align=right>1530</td><td align=right>1807</td></tr> |
<td align=right>1</td><td align=right>25</td><td align=right>72</td><td align=right>142</td><td align=right>235</td><td align=right>351</td><td align=right>490</td><td align=right>652</td><td align=right>837</td><td align=right>1045</td><td align=right>1276</td><td align=right>1530</td><td align=right>1807</td></tr> |
||
<tr><td align=center>26- |
<tr><td align=center>26-கோணம்</td><td align=center>½<i>n</i>(24<i>n</i> - 22)</td> |
||
<td align=right>1</td><td align=right>26</td><td align=right>75</td><td align=right>148</td><td align=right>245</td><td align=right>366</td><td align=right>511</td><td align=right>680</td><td align=right>873</td><td align=right>1090</td><td align=right>1331</td><td align=right>1596</td><td align=right>1885</td></tr> |
<td align=right>1</td><td align=right>26</td><td align=right>75</td><td align=right>148</td><td align=right>245</td><td align=right>366</td><td align=right>511</td><td align=right>680</td><td align=right>873</td><td align=right>1090</td><td align=right>1331</td><td align=right>1596</td><td align=right>1885</td></tr> |
||
<tr><td align=center>27- |
<tr><td align=center>27-கோணம்</td><td align=center>½<i>n</i>(25<i>n</i> - 23)</td> |
||
<td align=right>1</td><td align=right>27</td><td align=right>78</td><td align=right>154</td><td align=right>255</td><td align=right>381</td><td align=right>532</td><td align=right>708</td><td align=right>909</td><td align=right>1135</td><td align=right>1386</td><td align=right>1662</td><td align=right>1963</td></tr> |
<td align=right>1</td><td align=right>27</td><td align=right>78</td><td align=right>154</td><td align=right>255</td><td align=right>381</td><td align=right>532</td><td align=right>708</td><td align=right>909</td><td align=right>1135</td><td align=right>1386</td><td align=right>1662</td><td align=right>1963</td></tr> |
||
<tr><td align=center>28- |
<tr><td align=center>28-கோணம்</td><td align=center>½<i>n</i>(26<i>n</i> - 24)</td> |
||
<td align=right>1</td><td align=right>28</td><td align=right>81</td><td align=right>160</td><td align=right>265</td><td align=right>396</td><td align=right>553</td><td align=right>736</td><td align=right>945</td><td align=right>1180</td><td align=right>1441</td><td align=right>1728</td><td align=right>2041</td></tr> |
<td align=right>1</td><td align=right>28</td><td align=right>81</td><td align=right>160</td><td align=right>265</td><td align=right>396</td><td align=right>553</td><td align=right>736</td><td align=right>945</td><td align=right>1180</td><td align=right>1441</td><td align=right>1728</td><td align=right>2041</td></tr> |
||
<tr><td align=center>29- |
<tr><td align=center>29-கோணம்</td><td align=center>½<i>n</i>(27<i>n</i> - 25)</td> |
||
<td align=right>1</td><td align=right>29</td><td align=right>84</td><td align=right>166</td><td align=right>275</td><td align=right>411</td><td align=right>574</td><td align=right>764</td><td align=right>981</td><td align=right>1225</td><td align=right>1496</td><td align=right>1794</td><td align=right>2119</td></tr> |
<td align=right>1</td><td align=right>29</td><td align=right>84</td><td align=right>166</td><td align=right>275</td><td align=right>411</td><td align=right>574</td><td align=right>764</td><td align=right>981</td><td align=right>1225</td><td align=right>1496</td><td align=right>1794</td><td align=right>2119</td></tr> |
||
<tr><td align=center>30- |
<tr><td align=center>30-கோணம்</td><td align=center>½<i>n</i>(28<i>n</i> - 26)</td> |
||
<td align=right>1</td><td align=right>30</td><td align=right>87</td><td align=right>172</td><td align=right>285</td><td align=right>426</td><td align=right>595</td><td align=right>792</td><td align=right>1017</td><td align=right>1270</td><td align=right>1551</td><td align=right>1860</td><td align=right>2197</td></tr> |
<td align=right>1</td><td align=right>30</td><td align=right>87</td><td align=right>172</td><td align=right>285</td><td align=right>426</td><td align=right>595</td><td align=right>792</td><td align=right>1017</td><td align=right>1270</td><td align=right>1551</td><td align=right>1860</td><td align=right>2197</td></tr> |
||
</table> |
</table> |
20:54, 25 பெப்பிரவரி 2004 இல் நிலவும் திருத்தம்
கணிதத்தில், ஒரு பல்கோண எண் என்பது, ஒழுங்கான பல்கோண வடிவில் ஒழுங்குபடுத்தக்கூடிய ஒரு எண்ணாகும். பண்டைய கணிதவியலாளர்கள், எண்கள் கற்கள், விதைகள் போன்றவற்றால் குறிக்கப்படும்போது, அவற்றைக் குறிக்கப்பட்ட விதங்களில் ஒழுங்குபடுத்த முடியும் எனக் கண்டறிந்தனர். எடுத்துக்காட்டாக எண் 10 ஐ, ஒரு முக்கோணமாக ஒழுங்குபடுத்தமுடியும் (முக்கோண எண் பார்க்கவும்):
x x x x x x x x x x
ஆனால் 10 ஐ ஒரு சதுரமாக அடுக்கமுடியாது. எண் 9 ஐ அவ்வாறு அடுக்கமுடியும். (சதுர எண் பார்க்கவும்):
x x x x x x x x x
36 போன்ற சில எண்களைச் சதுரமாகவும், முக்கோணமாகவும் அடுக்கமுடியும். ( முக்கோண சதுர எண்):
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
The method for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as +.
முக்கோண எண்கள்
1:
+ x
3:
x x + + x x
6:
x x x x x x + + + x x x
10:
x x x x x x x x x x x x + + + + x x x x
சதுர எண்கள்
1:
+ x
4:
x + x x + + x x
9:
x x + x x x x x + x x x + + + x x x
16:
x x x + x x x x x x x + x x x x x x x + x x x x + + + + x x x x
ஐங்கோணி, அறுகோணி போன்ற கூடிய பக்கங்களைக்கொண்ட பல்கோணிகளும் புள்ளிகளின் அடுக்காகக் காட்டப்படக்கூடியவை. (convention படி, எத்தனை பக்கங்கொண்ட பல்கோணிக்கும் 1 முதலாவது பல்கோண எண்ணாகும்).
ஐங்கோண எண்கள்:
1:
+ x
5:
x x + + x x + + x x
12:
x x x x x x + x x + x x x x + + x x + + + x x x
22:
x x x x x x x x x x x x x x + x x + x x x x + x x x + x x x x x + + x x + + + + x x x x
35:
x x x x x x x x x x x x x x x x x x x x x x + x x x x x + x x x x x x x + x x + x x x x + x x x x + x x x x x x + + x x + + + + + x x x x x
Hexagonal numbers
1:
x
6:
x x + + x x + + x x + x
15:
x x x x x x + x x + x x x x + x + x x x + + x x + + x x + x
28:
x x x x x x x x x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
45:
x x x x x x x x x x x x x x x x x x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
66: (இது முக்கோண எண்ணும், ஒரு sphenic number உமாகும்.)
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x + x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
91:
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + x x x x x x x + x x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x x + x x x x x x x x + x x x x x + x x x x x x x + x x x x + x x x x x x + x x x x + x x x x x x + x x x + x x x x x + x x + x x x x + x x + x x x x + x + x x x + + x x + + x x + x
s என்பது பல்கோணியின் பக்க எண்ணிக்கையும், the formula nஆவது sthe nth s-polygonal number is ½n((s-2)n - (4-s)).
பெயர் | Formula | n=1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
முக்கோணம் | ½n(1n + 1) | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 |
சதுரம் | ½n(2n - 0) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 |
ஐங்கோணம் | ½n(3n - 1) | 1 | 5 | 12 | 22 | 35 | 51 | 70 | 92 | 117 | 145 | 176 | 210 | 247 |
அறுகோணம் | ½n(4n - 2) | 1 | 6 | 15 | 28 | 45 | 66 | 91 | 120 | 153 | 190 | 231 | 276 | 325 |
எழுகோணம் | ½n(5n - 3) | 1 | 7 | 18 | 34 | 55 | 81 | 112 | 148 | 189 | 235 | 286 | 342 | 403 |
எண்கோணம் | ½n(6n - 4) | 1 | 8 | 21 | 40 | 65 | 96 | 133 | 176 | 225 | 280 | 341 | 408 | 481 |
நவகோணம் | ½n(7n - 5) | 1 | 9 | 24 | 46 | 75 | 111 | 154 | 204 | 261 | 325 | 396 | 474 | 559 |
தசகோணம் | ½n(8n - 6) | 1 | 10 | 27 | 52 | 85 | 126 | 175 | 232 | 297 | 370 | 451 | 540 | 637 |
11-கோணம் | ½n(9n - 7) | 1 | 11 | 30 | 58 | 95 | 141 | 196 | 260 | 333 | 415 | 506 | 606 | 715 |
12-கோணம் | ½n(10n - 8) | 1 | 12 | 33 | 64 | 105 | 156 | 217 | 288 | 369 | 460 | 561 | 672 | 793 |
13-கோணம் | ½n(11n - 9) | 1 | 13 | 36 | 70 | 115 | 171 | 238 | 316 | 405 | 505 | 616 | 738 | 871 |
14-கோணம் | ½n(12n - 10) | 1 | 14 | 39 | 76 | 125 | 186 | 259 | 344 | 441 | 550 | 671 | 804 | 949 |
15-கோணம் | ½n(13n - 11) | 1 | 15 | 42 | 82 | 135 | 201 | 280 | 372 | 477 | 595 | 726 | 870 | 1027 |
16-கோணம் | ½n(14n - 12) | 1 | 16 | 45 | 88 | 145 | 216 | 301 | 400 | 513 | 640 | 781 | 936 | 1105 |
17-கோணம் | ½n(15n - 13) | 1 | 17 | 48 | 94 | 155 | 231 | 322 | 428 | 549 | 685 | 836 | 1002 | 1183 |
18-கோணம் | ½n(16n - 14) | 1 | 18 | 51 | 100 | 165 | 246 | 343 | 456 | 585 | 730 | 891 | 1068 | 1261 |
19-கோணம் | ½n(17n - 15) | 1 | 19 | 54 | 106 | 175 | 261 | 364 | 484 | 621 | 775 | 946 | 1134 | 1339 |
20-கோணம் | ½n(18n - 16) | 1 | 20 | 57 | 112 | 185 | 276 | 385 | 512 | 657 | 820 | 1001 | 1200 | 1417 |
21-கோணம் | ½n(19n - 17) | 1 | 21 | 60 | 118 | 195 | 291 | 406 | 540 | 693 | 865 | 1056 | 1266 | 1495 |
22-கோணம் | ½n(20n - 18) | 1 | 22 | 63 | 124 | 205 | 306 | 427 | 568 | 729 | 910 | 1111 | 1332 | 1573 |
23-கோணம் | ½n(21n - 19) | 1 | 23 | 66 | 130 | 215 | 321 | 448 | 596 | 765 | 955 | 1166 | 1398 | 1651 |
24-கோணம் | ½n(22n - 20) | 1 | 24 | 69 | 136 | 225 | 336 | 469 | 624 | 801 | 1000 | 1221 | 1464 | 1729 |
25-கோணம் | ½n(23n - 21) | 1 | 25 | 72 | 142 | 235 | 351 | 490 | 652 | 837 | 1045 | 1276 | 1530 | 1807 |
26-கோணம் | ½n(24n - 22) | 1 | 26 | 75 | 148 | 245 | 366 | 511 | 680 | 873 | 1090 | 1331 | 1596 | 1885 |
27-கோணம் | ½n(25n - 23) | 1 | 27 | 78 | 154 | 255 | 381 | 532 | 708 | 909 | 1135 | 1386 | 1662 | 1963 |
28-கோணம் | ½n(26n - 24) | 1 | 28 | 81 | 160 | 265 | 396 | 553 | 736 | 945 | 1180 | 1441 | 1728 | 2041 |
29-கோணம் | ½n(27n - 25) | 1 | 29 | 84 | 166 | 275 | 411 | 574 | 764 | 981 | 1225 | 1496 | 1794 | 2119 |
30-கோணம் | ½n(28n - 26) | 1 | 30 | 87 | 172 | 285 | 426 | 595 | 792 | 1017 | 1270 | 1551 | 1860 | 2197 |
References
- The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [ISBN 0140261494].