Page 4 of 8
Carréphylic classes: n2+1
Here are the first 10 carréphylic numbers of the form n2+1 with the usual data. The qt-blocks in the middle split the sections in a left part below and right part above the root-value. For all q/t fractions v=-1.
A fraction for which v=-1 is always a q/t fraction, always in the middle, and constitutes the only occasion where the denominator of the next fraction in a section surpasses the numerator of the previous one, in carréphylic as well as carréphocic numbers.
To understand the nature of the exception one would have to understand the nature of the rule, which is that the next denominator at most equals the previous numerator, and that only in the sp-blocks. The explanations of both rule and exception have till now eluded me.
√2
1 | 0 | 1 | 3 | 4 | 7 | 17 | 24 | 41 | 99 | 140 | 239 | 577 | 816 | 1393 | 3363 | 4756 | 8119 | 19601 | 27720 | 47321 | 114243 | 161564 | 275807 | 665857 | 941664 | 1607521 | 3880899 | 5488420 | 9369319 | 22619537 | 31988856 | 54608393 | 131836323 | 186444716 | ... |
0 | 1 | 1 | 2 | 3 | 5 | 12 | 17 | 29 | 70 | 99 | 169 | 408 | 577 | 985 | 2378 | 3363 | 5741 | 13860 | 19601 | 33461 | 80782 | 114243 | 195025 | 470832 | 665857 | 1136689 | 2744210 | 3880899 | 6625109 | 15994428 | 22619537 | 38613965 | 93222358 | 131836323 | ... |
√5
1 | 0 | 1 | 2 | 7 | 9 | 20 | 29 | 38 | 123 | 161 | 360 | 521 | 682 | 2207 | 2889 | 6460 | 9349 | 12238 | 39603 | 51841 | 115920 | 167761 | 219602 | 710647 | 930249 | 2080100 | 3010349 | 3940598 | 12752043 | 16692641 | 37325880 | 54018521 | 70711162 | 228826127 | 299537289 | 669785740 | ... |
0 | 1 | 1 | 1 | 3 | 4 | 9 | 13 | 17 | 55 | 72 | 161 | 233 | 305 | 987 | 1292 | 2889 | 4181 | 5473 | 17711 | 23184 | 51841 | 75025 | 98209 | 317811 | 416020 | 930249 | 1346269 | 1762289 | 5702887 | 7465176 | 16692641 | 24157817 | 31622993 | 102334155 | 133957148 | 299537289 | ... |
√10
1 | 0 | 1 | 2 | 3 | 13 | 16 | 19 | 60 | 79 | 98 | 117 | 487 | 604 | 721 | 2280 | 3001 | 3722 | 4443 | 18493 | 22936 | 27379 | 86580 | 113959 | 141338 | 168717 | 702247 | 870964 | 1039681 | 3287760 | 4327441 | 5367122 | 6406803 | 26666893 | 33073696 | 39480499 | 124848300 | ... |
0 | 1 | 1 | 1 | 1 | 4 | 5 | 6 | 19 | 25 | 31 | 37 | 154 | 191 | 228 | 721 | 949 | 1177 | 1405 | 5848 | 7253 | 8658 | 27379 | 36037 | 44695 | 53353 | 222070 | 275423 | 328776 | 1039681 | 1368457 | 1697233 | 2026009 | 8432812 | 10458821 | 12484830 | 39480499 | ... |
√17
1 | 0 | 1 | 2 | 3 | 4 | 21 | 25 | 29 | 33 | 136 | 169 | 202 | 235 | 268 | 1373 | 1641 | 1909 | 2177 | 8976 | 11153 | 13330 | 15507 | 17684 | 90597 | 108281 | 125965 | 143649 | 592280 | 735929 | 879578 | 1023227 | 1166876 | 5978029 | 7144905 | 8311781 | 9478657 | 39081504 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 5 | 6 | 7 | 8 | 33 | 41 | 49 | 57 | 65 | 333 | 398 | 463 | 528 | 2177 | 2705 | 3233 | 3761 | 4289 | 21973 | 26262 | 30551 | 34840 | 143649 | 178489 | 213329 | 248169 | 283009 | 1449885 | 1732894 | 2015903 | 2298912 | 9478657 | ... |
√26
1 | 0 | 1 | 2 | 3 | 4 | 5 | 31 | 36 | 41 | 46 | 51 | 260 | 311 | 362 | 413 | 464 | 515 | 3141 | 3656 | 4171 | 4686 | 5201 | 26520 | 31721 | 36922 | 42123 | 47324 | 52525 | 320351 | 372876 | 425401 | 477926 | 530451 | 2704780 | 3235231 | 3765682 | 4296133 | 4826584 | 5357035 | 32672661 | 38029696 | 43386731 | 48743766 | 54100801 | 275861040 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 7 | 8 | 9 | 10 | 51 | 61 | 71 | 81 | 91 | 101 | 616 | 717 | 818 | 919 | 1020 | 5201 | 6221 | 7241 | 8261 | 9281 | 10301 | 62826 | 73127 | 83428 | 93729 | 104030 | 530451 | 634481 | 738511 | 842541 | 946571 | 1050601 | 6407636 | 7458237 | 8508838 | 9559439 | 10610040 | 54100801 | ... |
√37
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 43 | 49 | 55 | 61 | 67 | 73 | 444 | 517 | 590 | 663 | 736 | 809 | 882 | 6247 | 7129 | 8011 | 8893 | 9775 | 10657 | 64824 | 75481 | 86138 | 96795 | 107452 | 118109 | 128766 | 912019 | 1040785 | 1169551 | 1298317 | 1427083 | 1555849 | 9463860 | 11019709 | 12575558 | 14131407 | 15687256 | 17243105 | 18798954 | 133148527 | 151947481 | 170746435 | 189545389 | 208344343 | 227143297 | 1381658736 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 8 | 9 | 10 | 11 | 12 | 73 | 85 | 97 | 109 | 121 | 133 | 145 | 1027 | 1172 | 1317 | 1462 | 1607 | 1752 | 10657 | 12409 | 14161 | 15913 | 17665 | 19417 | 21169 | 149935 | 171104 | 192273 | 213442 | 234611 | 255780 | 1555849 | 1811629 | 2067409 | 2323189 | 2578969 | 2834749 | 3090529 | 21889483 | 24980012 | 28070541 | 31161070 | 34251599 | 37342128 | 227143297 | ... |
√50
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 57 | 64 | 71 | 78 | 85 | 92 | 99 | 700 | 799 | 898 | 997 | 1096 | 1195 | 1294 | 1393 | 11243 | 12636 | 14029 | 15422 | 16815 | 18208 | 19601 | 138600 | 158201 | 177802 | 197403 | 217004 | 236605 | 256206 | 275807 | 2226057 | 2501864 | 2777671 | 3053478 | 3329285 | 3605092 | 3880899 | 27442100 | 31322999 | 35203898 | 39084797 | 42965696 | 46846595 | 50727494 | 54608393 | 440748043 | 495356436 | 549964829 | 604573222 | 659181615 | 713790008 | 768398401 | 5433397200 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 99 | 113 | 127 | 141 | 155 | 169 | 183 | 197 | 1590 | 1787 | 1984 | 2181 | 2378 | 2575 | 2772 | 19601 | 22373 | 25145 | 27917 | 30689 | 33461 | 36233 | 39005 | 314812 | 353817 | 392822 | 431827 | 470832 | 509837 | 548842 | 3880899 | 4429741 | 4978583 | 5527425 | 6076267 | 6625109 | 7173951 | 7722793 | 62331186 | 70053979 | 77776772 | 85499565 | 93222358 | 100945151 | 108667944 | 768398401 | ... |
√65
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 73 | 81 | 89 | 97 | 105 | 113 | 121 | 129 | 1040 | 1169 | 1298 | 1427 | 1556 | 1685 | 1814 | 1943 | 2072 | 18777 | 20849 | 22921 | 24993 | 27065 | 29137 | 31209 | 33281 | 268320 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 129 | 145 | 161 | 177 | 193 | 209 | 225 | 241 | 257 | 2329 | 2586 | 2843 | 3100 | 3357 | 3614 | 3871 | 4128 | 33281 | ... |
√82
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 91 | 100 | 109 | 118 | 127 | 136 | 145 | 154 | 163 | 1476 | 1639 | 1802 | 1965 | 2128 | 2291 | 2454 | 2617 | 2780 | 2943 | 29593 | 32536 | 35479 | 38422 | 41365 | 44308 | 47251 | 50194 | 53137 | 481176 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 163 | 181 | 199 | 217 | 235 | 253 | 271 | 289 | 307 | 325 | 3268 | 3593 | 3918 | 4243 | 4568 | 4893 | 5218 | 5543 | 5868 | 53137 | ... |
√101
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 111 | 121 | 131 | 141 | 151 | 161 | 171 | 181 | 191 | 201 | 2020 | 2221 | 2422 | 2623 | 2824 | 3025 | 3226 | 3427 | 3628 | 3829 | 4030 | 44531 | 48561 | 52591 | 56621 | 60651 | 64681 | 68711 | 72741 | 76771 | 80801 | 812040 | ... |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 201 | 221 | 241 | 261 | 281 | 301 | 321 | 341 | 361 | 381 | 401 | 4431 | 4832 | 5233 | 5634 | 6035 | 6436 | 6837 | 7238 | 7639 | 8040 | 80801 | ... |