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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 ...