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Certain classses of numbers are inherently carréphylic. Here are the five that come with every square and two more that come with even squares:

They show convincingly that members of the same class have a similar approach profile.

Carréphylic classes: n2-2

Here are the first 10 carréphylic numbers of the form n2-2 with the usual data.
Note that all fractions except the s/p- and t/q-fractions are below the root-value.
A nice tq-pattern over even and odd squares.
 For all t/q fractions involved: v = 2 For q/t-fractions 'q' follows: U(1,7,4)2 1, 7, 17, 31, 49, 71, ... while 'v' follows its negative: U(-1,-7,-4)2 -1, -7, -17, -31, -49, -71, ...

√7
 1 0 1 2 3 5 8 21 29 37 45 82 127 336 463 590 717 1307 2024 5355 7379 9403 11427 20830 32257 85344 117601 149858 182115 331973 514088 1360149 1874237 2388325 2902413 5290738 8193151 21677040 29870191 38063342 46256493 84319835 130576328 345472491 ... 0 1 1 1 1 2 3 8 11 14 17 31 48 127 175 223 271 494 765 2024 2789 3554 4319 7873 12192 32257 44449 56641 68833 125474 194307 514088 708395 902702 1097009 1999711 3096720 8193151 11289871 14386591 17483311 31869902 49353213 130576328 ...

√14
 1 0 1 2 3 4 7 11 15 56 71 86 101 116 217 333 449 1680 2129 2578 3027 3476 6503 9979 13455 50344 63799 77254 90709 104164 194873 299037 403201 1508640 1911841 2315042 2718243 3121444 5839687 8961131 12082575 45208856 ... 0 1 1 1 1 1 2 3 4 15 19 23 27 31 58 89 120 449 569 689 809 929 1738 2667 3596 13455 17051 20647 24243 27839 52082 79921 107760 403201 510961 618721 726481 834241 1560722 2394963 3229204 12082575 ...

√23
 1 0 1 2 3 4 5 14 19 24 115 139 163 187 211 235 681 916 1151 5520 6671 7822 8973 10124 11275 32674 43949 55224 264845 320069 375293 430517 485741 540965 1567671 2108636 2649601 12707040 15356641 18006242 20655843 23305444 25955045 75215534 101170579 127125624 609673075 ... 0 1 1 1 1 1 1 3 4 5 24 29 34 39 44 49 142 191 240 1151 1391 1631 1871 2111 2351 6813 9164 11515 55224 66739 375293 89769 101284 112799 326882 439681 552480 2649601 3202081 3754561 4307041 4859521 5412001 15683523 21095524 26507525 127125624 ...

√34
 1 0 1 2 3 4 5 6 17 23 29 35 204 239 274 309 344 379 414 1207 1621 2035 2449 14280 16729 19178 21627 24076 26525 28974 84473 113447 142421 171395 999396 1170791 1342186 1513581 1684976 1856371 2027766 5911903 7939669 9967435 11995201 69943440 ... 0 1 1 1 1 1 1 1 3 4 5 6 35 41 47 53 59 65 71 207 278 349 420 2449 2869 3289 3709 4129 4549 4969 14487 19456 24425 29394 171395 200789 230183 259577 288971 318365 347759 1013883 1361642 1709401 2057160 11995201 ...

√47
 1 0 1 2 3 4 5 6 7 27 34 41 48 329 377 425 473 521 569 617 665 2612 3277 3942 4607 31584 36191 40798 45405 50012 54619 59226 63833 250725 314558 378391 442224 3031735 3473959 3916183 4358407 4800631 5242855 5685079 6127303 24066988 30194291 36321594 42448897 291014976 ... 0 1 1 1 1 1 1 1 1 4 5 6 7 48 55 62 69 76 83 90 97 381 478 575 672 4607 5279 5951 6623 7295 7967 8639 9311 36572 45883 55194 64505 442224 506729 571234 635739 700244 764749 829254 893759 3510531 4404290 5298049 6191808 42448897 ...

√62
 1 0 1 2 3 4 5 6 7 8 31 39 47 55 63 496 559 622 685 748 811 874 937 1000 3937 4937 5937 6937 7937 62496 70433 78370 86307 94244 102181 110118 118055 125992 490631 622023 748015 874007 999999 7874000 ... 0 1 1 1 1 1 1 1 1 1 4 5 6 7 8 63 71 79 87 95 103 111 119 127 500 627 754 881 1008 7937 8945 9953 10961 11969 12977 13985 14993 16001 62996 78997 94998 110999 127000 999999 ...

√79
 1 0 1 2 3 4 5 6 7 8 9 44 53 62 71 80 711 791 871 951 1031 1111 1191 1271 1351 1431 7075 8506 9937 11368 12799 113760 ... 0 1 1 1 1 1 1 1 1 1 1 5 6 7 8 9 80 89 98 107 116 125 134 143 152 161 796 957 1118 1279 1440 12799 ...

√98
 1 0 1 2 3 4 5 6 7 8 9 10 49 59 69 79 89 99 980 1079 1178 1277 1376 1475 1574 1673 1772 1871 1970 9751 11721 13691 15661 17631 19601 194040 ... 0 1 1 1 1 1 1 1 1 1 1 1 5 6 7 8 9 10 99 109 119 129 139 149 159 169 179 189 199 985 1184 1383 1582 1781 1980 19601 ...

√119
 1 0 1 2 3 4 5 6 7 8 9 10 11 65 76 87 98 109 120 1309 1429 1549 1669 1789 1909 2029 2149 2269 2389 2509 2629 15654 18283 20912 23541 26170 28799 314160 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 6 7 8 9 10 11 120 131 142 153 164 175 186 197 208 219 230 241 1435 1676 1917 2158 2399 2640 28799 ...

√142
 1 0 1 2 3 4 5 6 7 8 9 10 11 12 71 83 95 107 119 131 143 1704 1847 1990 2133 2276 2419 2562 2705 2848 2991 3134 3277 3420 20377 23797 27217 30637 34057 37477 40897 487344 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 1 6 7 8 9 10 11 12 143 155 167 179 191 203 215 227 239 251 263 275 287 1710 1997 2284 2571 2858 3145 3432 40897 ...

Carréphylic classes: n2-1

Here are the first 10 carréphylic numbers of the form n2-1 with the usual data.
For numbers 'n' that are one less than a square, or N2-n = 1, the formula gives the sp-fraction and factor of the base-2 accelleration of the series.
The first fraction is N/1, F=2N, because obviously N2-n*12 = 1 is a solution of the diophantine equation.
Note that the q/t fractions immediately precede the sp-blocks, that their v-value decreases with steps of 2 over 'n' and that all approximations except the s/p-fractions are below the root-value.
Compare n2+n.

√3
 1 0 1 2 3 5 7 12 19 26 45 71 97 168 265 362 627 989 1351 2340 3691 5042 8733 13775 18817 32592 51409 70226 121635 191861 262087 453948 716035 978122 1694157 ... 0 1 1 1 2 3 4 7 11 15 26 41 56 97 153 209 362 571 780 1351 2131 2911 5042 7953 10864 18817 29681 40545 70226 110771 151316 262087 413403 564719 978122 ...

√8
 1 0 1 2 3 8 11 14 17 48 65 82 99 280 379 478 577 1632 2209 2786 3363 9512 12875 16238 19601 55440 75041 94642 114243 323128 437371 551614 665857 1883328 2549185 3215042 3880899 10976840 14857739 18738638 22619537 63977712 ... 0 1 1 1 1 3 4 5 6 17 23 29 35 99 134 169 204 577 781 985 1189 3363 4552 5741 6930 19601 26531 33461 40391 114243 154634 195025 235416 665857 901273 1136689 1372105 3880899 5253004 6625109 7997214 22619537 ...

√15
 1 0 1 2 3 4 15 19 23 27 31 120 151 182 213 244 945 1189 1433 1677 1921 7440 9361 11282 13203 15124 58575 73699 88823 103947 119071 461160 580231 699302 818373 937444 3630705 4568149 5505593 6443037 7380481 28584480 ... 0 1 1 1 1 1 4 5 6 7 8 31 39 47 55 63 244 307 370 433 496 1921 2417 2913 3409 3905 15124 19029 22934 26839 30744 119071 149815 180559 211303 242047 937444 1179491 1421538 1663585 1905632 7380481 ...

√24
 1 0 1 2 3 4 5 24 29 34 39 44 49 240 289 338 387 436 485 2376 2861 3346 3831 4316 4801 23520 28321 33122 37923 42724 47525 232824 280349 327874 375399 422924 470449 2304720 2775169 3245618 3716067 4186516 4656965 22814376 27471341 32128306 36785271 41442236 46099201 225839040 ... 0 1 1 1 1 1 1 5 6 7 8 9 10 49 59 69 79 89 99 485 584 683 782 881 980 4801 5781 6761 7741 8721 9701 47525 57226 66927 76628 86329 96030 470449 566479 662509 758539 854569 950599 4656965 5607564 6558163 7508762 8459361 9409960 46099201 ...

√35
 1 0 1 2 3 4 5 6 35 41 47 53 59 65 71 420 491 562 633 704 775 846 5005 5851 6697 7543 8389 9235 10081 59640 69721 79802 89883 99964 110045 120126 710675 830801 950927 1071053 1191179 1311305 1431431 8468460 9899891 11331322 12762753 14194184 15625615 17057046 100910845 ... 0 1 1 1 1 1 1 1 6 7 8 9 10 11 12 71 83 95 107 119 131 143 846 989 1132 1275 1418 1561 1704 10081 11785 13489 15193 16897 18601 20305 120126 140431 160736 181041 201346 221651 241956 1431431 1673387 1915343 2157299 2399255 2641211 2883167 17057046 ...

√48
 1 0 1 2 3 4 5 6 7 48 55 62 69 76 83 90 97 672 769 866 963 1060 1157 1254 1351 9360 10711 12062 13413 14764 16115 17466 18817 130368 149185 168002 186819 205636 224453 243270 262087 1815792 2077879 2339966 2602053 2864140 3126227 3388314 3650401 25290720 ... 0 1 1 1 1 1 1 1 1 7 8 9 10 11 12 13 14 97 111 125 139 153 167 181 195 1351 1546 1741 1936 2131 2326 2521 2716 18817 21533 24249 26965 29681 32397 35113 37829 262087 299916 337745 375574 413403 451232 489061 526890 3650401 ...

√63
 1 0 1 2 3 4 5 6 7 8 63 71 79 87 95 103 111 119 127 1008 1135 1262 1389 1516 1643 1770 1897 2024 16065 18089 20113 22137 24161 26185 28209 30233 32257 256032 ... 0 1 1 1 1 1 1 1 1 1 8 9 10 11 12 13 14 15 16 127 143 159 175 191 207 223 239 255 2024 2279 2534 2789 3044 3299 3554 3809 4064 32257 ...

√80
 1 0 1 2 3 4 5 6 7 8 9 80 89 98 107 116 125 134 143 152 161 1440 1601 1762 1923 2084 2245 2406 2567 2728 2889 25840 28729 31618 34507 37396 40285 43174 46063 48952 51841 463680 515521 567362 619203 671044 722885 774726 826567 878408 930249 8320400 9250649 10180898 11111147 12041396 12971645 13901894 14832143 15762392 16692641 149303520 165996161 182688802 199381443 216074084 232766725 249459366 266152007 282844648 299537289 2679142960 ... 0 1 1 1 1 1 1 1 1 1 1 9 10 11 12 13 14 15 16 17 18 161 179 197 215 233 251 269 287 305 323 2889 3212 3535 3858 4181 4504 4827 5150 5473 5796 51841 57637 63433 69229 75025 80821 86617 92413 98209 104005 930249 1034254 1138259 1242264 1346269 1450274 1554279 1658284 1762289 1866294 16692641 18558935 20425229 22291523 24157817 26024111 27890405 29756699 31622993 33489287 299537289 ...

√99
 1 0 1 2 3 4 5 6 7 8 9 10 99 109 119 129 139 149 159 169 179 189 199 1980 2179 2378 2577 2776 2975 3174 3373 3572 3771 3970 39501 ... 0 1 1 1 1 1 1 1 1 1 1 1 10 11 12 13 14 15 16 17 18 19 20 199 219 239 259 279 299 319 339 359 379 399 3970 ...

√120
 1 0 1 2 3 4 5 6 7 8 9 10 11 120 131 142 153 164 175 186 197 208 219 230 241 2640 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 11 12 13 14 15 16 17 18 19 20 21 22 241 ...

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 ...

Carréphylic classes: n2+2

Nice and orderly. For all q/t-fractions, v = -2, placing the it to the right of the middle of the section but keeping up the 'v = -1 division' of fractions to the left and right being below and above the root-value respectively.

√6
 1 0 1 2 5 12 17 22 49 120 169 218 485 1188 1673 2158 4801 11760 16561 21362 47525 116412 163937 211462 470449 1152360 1622809 2093258 4656965 11407188 ... 0 1 1 1 2 5 7 9 20 49 69 89 198 485 683 881 1960 4801 6761 8721 19402 47525 66927 86329 192060 470449 662509 854569 1901198 4656965 ...

√11
 1 0 1 2 3 7 10 33 43 53 63 136 199 660 859 1058 1257 2713 3970 13167 17137 21107 25077 54124 79201 262680 341881 421082 500283 1079767 1580050 5240433 6820483 8400533 9980583 21541216 31521799 104545980 ... 0 1 1 1 1 2 3 10 13 16 19 41 60 199 259 319 379 818 1197 3970 5167 6364 7561 16319 23880 79201 103081 126961 150841 325562 476403 1580050 2056453 2532856 3009259 6494921 9504180 31521799 ...

√18
 1 0 1 2 3 4 13 17 72 89 106 123 140 437 577 2448 3025 3602 4179 4756 14845 19601 83160 102761 122362 141963 161564 504293 665857 2824992 3490849 4156706 4822563 5488420 17131117 22619537 95966568 ... 0 1 1 1 1 1 3 4 17 21 25 29 33 103 136 577 713 849 985 1121 3499 4620 19601 24221 28841 33461 38081 118863 156944 665857 822801 979745 1136689 1293633 4037843 5331476 22619537 ...

√27
 1 0 1 2 3 4 5 16 21 26 135 161 187 213 239 265 821 1086 1351 7020 8371 9722 11073 12424 13775 42676 56451 70226 364905 435131 505357 575583 645809 716035 2218331 2934366 3650401 18968040 22618441 26268842 29919243 33569644 37220045 115310536 152530581 189750626 985973175 ... 0 1 1 1 1 1 1 3 4 5 26 31 36 41 46 51 158 209 260 1351 1611 1871 2131 2391 2651 8213 10864 13515 70226 83741 97256 110771 124286 137801 426918 564719 702520 3650401 4352921 5055441 5757961 6460481 7163001 22191523 29354524 36517525 189750626 ...

√38
 1 0 1 2 3 4 5 6 25 31 37 228 265 302 339 376 413 450 1837 2287 2737 16872 19609 22346 25083 27820 30557 33294 135913 169207 202501 1248300 1450801 1653302 1855803 2058304 2260805 2463306 10055725 12519031 14982337 179853744 107339665 122322002 137304339 152286676 167269013 182251350 743987737 926239087 1108490437 6833193972 ... 0 1 1 1 1 1 1 1 4 5 6 37 43 49 55 61 67 73 298 371 444 2737 3181 3625 4069 4513 4957 5401 22048 27449 32850 202501 235351 268201 301051 333901 366751 399601 1631254 2030855 2430456 14982337 17412793 19843249 22273705 24704161 27134617 29565073 120690748 150255821 179820894 1108490437 ...

√51
 1 0 1 2 3 4 5 6 7 29 36 43 50 357 407 457 507 557 607 657 707 2878 3585 4292 4999 35700 40699 45698 50697 55696 60695 65694 70693 287771 358464 429157 499850 3569643 4069493 4569343 5069193 5569043 6068893 6568743 7068593 28774222 35842815 42911408 49980001 356928600 ... 0 1 1 1 1 1 1 1 1 4 5 6 7 50 57 64 71 78 85 92 99 403 502 601 700 4999 5699 6399 7099 7799 8499 9199 9899 40296 50195 60094 69993 499850 569843 639836 709829 779822 849815 919808 989801 4029197 5018998 6008799 6998600 49980001 ...

√66
 1 0 1 2 3 4 5 6 7 8 41 49 57 65 528 593 658 723 788 853 918 983 1048 5305 6353 7401 8449 68640 ... 0 1 1 1 1 1 1 1 1 1 5 6 7 8 65 73 81 89 97 105 113 121 129 653 782 911 1040 8449 ...

√83
 1 0 1 2 3 4 5 6 7 8 9 46 55 64 73 82 747 829 911 993 1075 1157 1239 1321 1403 1485 7507 8992 10477 11962 13447 122508 ... 0 1 1 1 1 1 1 1 1 1 1 5 6 7 8 9 82 91 100 109 118 127 136 145 154 163 824 987 1150 1313 1476 13447 ...

√102
 1 0 1 2 3 4 5 6 7 8 9 10 61 71 81 91 101 1020 1121 1222 1323 1424 1525 1626 1727 1828 1929 2030 12281 14311 16341 18371 20401 206040 ... 0 1 1 1 1 1 1 1 1 1 1 1 6 7 8 9 10 101 111 121 131 141 151 161 171 181 191 201 1216 1417 1618 1819 2020 20401 ...

√123
 1 0 1 2 3 4 5 6 7 8 9 10 11 67 78 89 100 111 122 1353 1475 1597 1719 1841 1963 2085 2207 2329 2451 2573 2695 16292 18987 21682 24377 27072 29767 330132 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 6 7 8 9 10 11 122 133 144 155 166 177 188 199 210 221 232 243 1469 1712 1955 2198 2441 2684 29767 ...

Carréphylic classes: n2+n

Note that the q/t fractions immediately precede the sp-blocks, that their v-value decreases with steps of 1 over 'n' and that all approximations except the s/p-fractions are below the root-value.
Compare n2-1.

√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 ...

√6
 1 0 1 2 5 12 17 22 49 120 169 218 485 1188 1673 2158 4801 11760 16561 21362 47525 116412 163937 211462 470449 1152360 1622809 2093258 4656965 11407188 ... 0 1 1 1 2 5 7 9 20 49 69 89 198 485 683 881 1960 4801 6761 8721 19402 47525 66927 86329 192060 470449 662509 854569 1901198 4656965 ...

√12
 1 0 1 2 3 7 24 31 38 45 97 336 433 530 627 1351 4680 6031 7382 8733 18817 65184 84001 102818 121635 262087 907896 1169983 1432070 1694157 3650401 12645360 16295761 19946162 23596563 50843527 176127144 ... 0 1 1 1 1 2 7 9 11 13 28 97 125 153 181 390 1351 1741 2131 2521 5432 18817 24249 29681 35113 75658 262087 337745 413403 489061 1053780 3650401 4704181 5757961 6811741 14677262 50843527 ...

√20
 1 0 1 2 3 4 9 40 49 58 67 76 161 720 881 1042 1203 1364 2889 12920 15809 18698 21587 24476 51841 231840 283681 335522 387363 439204 930249 4160200 5090449 6020698 6950947 7881196 16692641 74651760 91344401 108037042 124729683 141422324 299537289 1339571480 1639108769 1938646058 2238183347 2537720636 5374978561 24037634880 ... 0 1 1 1 1 1 2 9 11 13 15 17 36 161 197 233 269 305 646 2889 3535 4181 4827 5473 11592 51841 63433 75025 86617 98209 208010 930249 1138259 1346269 1554279 1762289 3732588 16692641 20425229 24157817 27890405 31622993 66978574 299537289 366515863 433494437 500473011 567451585 1201881744 5374978561 ...

√30
 1 0 1 2 3 4 5 11 60 71 82 93 104 115 241 1320 1561 1802 2043 2284 2525 5291 28980 34271 39562 44853 50144 55435 116161 636240 752401 868562 984723 1100884 1217045 2550251 13968300 16518551 19068802 21619053 24169304 26719555 55989361 306666360 ... 0 1 1 1 1 1 1 2 11 13 15 17 19 21 44 241 285 329 373 417 461 966 5291 6257 7223 8189 9155 10121 21208 116161 137369 158577 179785 200993 222201 465610 2550251 3015861 3481471 3947081 4412691 4878301 10222212 55989361 ...

√42
 1 0 1 2 3 4 5 6 13 84 97 110 123 136 149 162 337 2184 2521 2858 3195 3532 3869 4206 8749 56700 65449 74198 82947 91696 100445 109194 227137 1472016 1699153 1926290 2153427 2380564 2607701 2834838 5896813 38215716 44112529 50009342 55906155 61802968 67699781 73596594 153090001 992136600 ... 0 1 1 1 1 1 1 1 2 13 15 17 19 21 23 25 52 337 389 441 493 545 597 649 1350 8749 10099 11449 12799 14149 15499 16849 35048 227137 262185 297233 332281 367329 402377 437425 909898 5896813 6806711 7716609 8626507 9536405 10446303 11356201 23622300 153090001 ...

√56
 1 0 1 2 3 4 5 6 7 15 112 127 142 157 172 187 202 217 449 3360 3809 4258 4707 5156 5605 6054 6503 13455 100688 114143 127598 141053 154508 167963 181418 194873 403201 3017280 3420481 3823682 4226883 4630084 5033285 5436486 5839687 12082575 90417712 102500287 114582862 126665437 138748012 150830587 162913162 174995737 362074049 2709514080 ... 0 1 1 1 1 1 1 1 1 2 15 17 19 21 23 25 27 29 60 449 509 569 629 689 749 809 869 1798 13455 15253 17051 18849 20647 22445 24243 26041 53880 403201 457081 510961 564841 618721 672601 726481 780361 1614602 12082575 13697177 15311779 16926381 18540983 20155585 21770187 23384789 48384180 362074049 ...

√72
 1 0 1 2 3 4 5 6 7 8 17 144 161 178 195 212 229 246 263 280 577 4896 5473 6050 6627 7204 7781 8358 8935 9512 19601 166320 185921 205522 225123 244724 264325 283926 303527 323128 665857 5649984 ... 0 1 1 1 1 1 1 1 1 1 2 17 19 21 23 25 27 29 31 33 68 577 645 713 781 849 917 985 1053 1121 2310 19601 21911 24221 26531 28841 31151 33461 35771 38081 78472 665857 ...

√90
 1 0 1 2 3 4 5 6 7 8 9 19 180 199 218 237 256 275 294 313 332 351 721 6840 ... 0 1 1 1 1 1 1 1 1 1 1 2 19 21 23 25 27 29 31 33 35 37 76 721 ...

√110
 1 0 1 2 3 4 5 6 7 8 9 10 21 220 241 262 283 304 325 346 367 388 409 430 881 9240 ... 0 1 1 1 1 1 1 1 1 1 1 1 2 21 23 25 27 29 31 33 35 37 39 41 84 881 ...

Carréphylic classes: n2-n/2 (for even n)

Here are the first 6 carréphylic numbers of the form n2-n/2 (for even n) with the usual data.
Root-3 is a 'small numbers exception' on the general pattern because it is also a member of n2-1 and shows the characteristic 'sp-block shift' of that class.
The first non-trivial ab-blocks (second for root-3) follow a simple pattern (4n-1)/4. The s/p- and t/q-fractions are the only ones above the root-value.
The values 'v' equal half the numerator of the first t/q fraction and the numerator of the first q/t fraction respectively.

√3
 1 0 1 2 3 5 7 12 19 26 45 71 97 168 265 362 627 989 1351 2340 3691 5042 8733 13775 18817 32592 51409 70226 121635 191861 262087 453948 716035 978122 1694157 ... 0 1 1 1 2 3 4 7 11 15 26 41 56 97 153 209 362 571 780 1351 2131 2911 5042 7953 10864 18817 29681 40545 70226 110771 151316 262087 413403 564719 978122 ...

√14
 1 0 1 2 3 4 7 11 15 56 71 86 101 116 217 333 449 1680 2129 2578 3027 3476 6503 9979 13455 50344 63799 77254 90709 104164 194873 299037 403201 1508640 1911841 2315042 2718243 3121444 5839687 8961131 12082575 45208856 ... 0 1 1 1 1 1 2 3 4 15 19 23 27 31 58 89 120 449 569 689 809 929 1738 2667 3596 13455 17051 20647 24243 27839 52082 79921 107760 403201 510961 618721 726481 834241 1560722 2394963 3229204 12082575 ...

√33
 1 0 1 2 3 4 5 6 11 17 23 132 155 178 201 224 247 270 517 787 1057 6072 7129 8186 9243 10300 11357 12414 23771 36185 48599 279180 327779 376378 424977 473576 522175 570774 1092949 1663723 2234497 12836208 15070705 17305202 19539699 21774196 24008693 26243190 50251883 76495073 102738263 590186388 ... 0 1 1 1 1 1 1 1 2 3 4 23 27 31 35 39 43 47 90 137 184 1057 1241 1425 1609 1793 1977 2161 4138 6299 8460 48599 57059 65519 73979 82439 90899 99359 190258 289617 388976 2234497 2623473 3012449 3401425 3790401 4179377 4568353 8747730 13316083 17884436 102738263 ...

√60
 1 0 1 2 3 4 5 6 7 8 15 23 31 240 271 302 333 364 395 426 457 488 945 1433 1921 14880 16801 18722 20643 22564 24485 26406 28327 30248 58575 88823 119071 922320 1041391 1160462 1279533 1398604 1517675 1636746 1755817 1874888 3630705 5505593 7380481 57168960 64549441 71929922 79310403 86690884 94071365 101451846 108832327 116212808 225045135 341257943 457470751 3543553200 ... 0 1 1 1 1 1 1 1 1 1 2 3 4 31 35 39 43 47 51 55 59 63 122 185 248 1921 2169 2417 2665 2913 3161 3409 3657 3905 7562 11467 15372 119071 134443 149815 165187 180559 195931 211303 226675 242047 468659 710769 952816 7380481 8333297 9286113 10238929 11191745 12144561 13097377 14050193 15003009 29045391 44056211 59059220 457470751 ...

√95
 1 0 1 2 3 4 5 6 7 8 9 10 19 29 39 380 419 458 497 536 575 614 653 692 731 770 1501 2271 3041 29640 ... 0 1 1 1 1 1 1 1 1 1 1 1 2 3 4 39 43 47 51 55 59 63 67 71 75 79 154 233 312 3041 ...

√138
 1 0 1 2 3 4 5 6 7 8 9 10 11 12 23 35 47 552 599 646 693 740 787 834 881 928 975 1022 1069 1116 2185 3301 4417 51888 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 47 51 55 59 63 67 71 75 79 83 87 91 95 186 281 376 4417 ...

Carréphylic classes: n2+n/2 (for even n)

Here are the first 6 carréphylic numbers of the form n2+n/2 (for even n) with the usual data.
The first non-trivial sp-blocks follow a simple pattern (4n+1)/n. The qt-blocks follow an even simpler one: n/1.
Their v-value equals -n/2.

√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 ...

√18
 1 0 1 2 3 4 13 17 72 89 106 123 140 437 577 2448 3025 3602 4179 4756 14845 19601 83160 102761 122362 141963 161564 504293 665857 2824992 3490849 4156706 4822563 5488420 17131117 22619537 95966568 ... 0 1 1 1 1 1 3 4 17 21 25 29 33 103 136 577 713 849 985 1121 3499 4620 19601 24221 28841 33461 38081 118863 156944 665857 822801 979745 1136689 1293633 4037843 5331476 22619537 ...

√39
 1 0 1 2 3 4 5 6 19 25 156 181 206 231 256 281 306 943 1249 7800 9049 10298 11547 12796 14045 15294 47131 62425 389844 452269 514694 577119 639544 701969 764394 2355607 3120001 19484400 22604401 25724402 28844403 31964404 35084405 38204406 117733219 155937625 973830156 ... 0 1 1 1 1 1 1 1 3 4 25 29 33 37 41 45 49 151 200 1249 1449 1649 1849 2049 2249 2449 7547 9996 62425 72421 82417 92413 102409 112405 122401 377199 499600 3120001 3619601 4119201 4618801 5118401 5618001 6117601 18852403 24970004 155937625 ...

√68
 1 0 1 2 3 4 5 6 7 8 25 33 272 305 338 371 404 437 470 503 536 1641 2177 17952 20129 22306 24483 26660 28837 31014 33191 35368 108281 143649 1184560 ... 0 1 1 1 1 1 1 1 1 1 3 4 33 37 41 45 49 53 57 61 65 199 264 2177 2441 2705 2969 3233 3497 3761 4025 4289 13131 17420 143649 ...

√105
 1 0 1 2 3 4 5 6 7 8 9 10 31 41 420 461 502 543 584 625 666 707 748 789 830 2531 3361 34440 ... 0 1 1 1 1 1 1 1 1 1 1 1 3 4 41 45 49 53 57 61 65 69 73 77 81 247 328 3361 ...

√150
 1 0 1 2 3 4 5 6 7 8 9 10 11 12 37 49 600 649 698 747 796 845 894 943 992 1041 1090 1139 1188 3613 4801 58800 ... 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3 4 49 53 57 61 65 69 73 77 81 85 89 93 97 295 392 4801 ...