Disclaimer: Tomas Oliveira e Silva kindly allowed us to post the data below, but neither he, nor us assume responsibility for any possible error.
Here is his email message, from Sept. 21, 2016
Dear Neil and Doron After reading arXiv:1609.05570v1 it became irresistible to conduct a quick search of lengthier Pisot sequences not satisfying a linear recurrence (but initially satisfying one). Here goes the highlights of what I found for 2 <= x <= 1998, x+2 <= y <= 2000, and r = 0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, and 1. For each r, the best (longest) 25 sequences are sent in the attached text file. I hope the data format is evident. The crown jewel is r=1/2, x=152, y=1422. The corresponding Pisot sequence fails the linear recurrence at the 547614th term! That's way longer than what Doron (or rather, SBE) found in his more limited search. Other cases worthy of notice are r=2/3, x=142, y=1924 (linear recurrence fails at 107470) and r=3/5, x=112, y=1220 (linear recurrence fails at 99210). Also worthy of notice is the case r=1/2, x=10, y=19+100*k, for k=0, 1, 2, 3, and so on. It appears that the corresponding Pisot sequences satisfy a 4th order linear recurrence (see attached file) for more and more initial terms. (y=4819, fails at 27319, y=4919 fails at 27865, and this trend appears to continue). Can it hold for arbitrary large k? It should be easy to settle this. If requested, I can provide the pari-gp script I used to conduct this initial search. In the next few days I am going to search up to 10000 (x and y), accepting linear recurrences of order up to 40. Would it be worthwhile to test rational values of r outside of the unit interval? It so happens that one of my side kicks (computers I have access to), is currently idle, and so this problem will keep it nicely occupied :-) Best regards, from a fan of the OEIS, Tomas Oliveira e Silva
/*
** search for Pisot sequences that almost satisfy a linear recurrence (arXiv:1609.05570v1)
**
** it looks like $x=10$, $y=19+10^2k$, $r=1/2$, gives rise to longer and longer fake linear
** recurrences, of the form a_n = (12+10k)a_{n-1} - (2+k)a_{n-2} + (10+8k)a_{n-3} - (6+5k)a_{n-4}
** this was tested for $0 \leq k < 100$. Does this pattern continue indefinitely?
** this phenomenon also appears for other values of $x$ and for $y$ in an arithmetic progression;
** for example, $x=23$ and $y=124+23^2k$, or $y=126+23^2k$, or $y=403+23^2k$, or $y=405+23^2k$
* also for $x=30$, $y=989+30^2k$?
** D. G. Cantor appears to have studied sequences of this form
**
** 25 longest fake sequences for r=0, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000022964456310 161 683 54068 13 [4,1,0,0,2,-1,0,0,0,-2,-2,0,-1]
** 1.000266808359533 481 1654 8279 14 [4,-2,0,1,0,-3,3,-2,0,1,1,-3,2,-1]
** 1.000091645618898 417 1400 7693 16 [3,0,4,0,1,-2,-1,-2,0,0,1,0,0,-1,-1,-1]
** 1.000142374097160 387 1296 6785 12 [3,1,1,-1,-1,-2,0,0,1,1,0,-2]
** 1.000245959654823 310 899 5790 18 [2,2,2,0,-1,-2,-2,-1,0,1,2,0,-1,-1,-1,0,0,1]
** 1.000268636529352 280 676 5629 16 [3,-2,2,-1,-1,0,0,-1,2,-1,1,-1,1,-1,1,-1]
** 1.000268636529352 676 1632 5628 16 [3,-2,2,-1,-1,0,0,-1,2,-1,1,-1,1,-1,1,-1]
** 1.000348262482604 41 276 5095 27 [7,-2,1,1,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1]
** 1.000348262482604 276 1857 5094 27 [7,-2,1,1,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1]
** 1.000416303546555 293 883 4633 13 [3,0,0,1,-2,0,1,-1,2,0,-2,0,-1]
** 1.000527961890823 383 1668 3054 11 [4,2,-2,0,0,2,-1,-2,1,-2,-1]
** 1.000479191568966 165 561 2602 10 [4,-2,0,-1,2,-1,0,0,1,-2]
** 1.000479191568966 561 1907 2601 10 [4,-2,0,-1,2,-1,0,0,1,-2]
** 1.001214184615256 464 1616 2560 18 [5,-7,8,-9,8,-5,3,-2,1,-1,1,-1,1,-1,1,-1,1,-1]
** 1.000470307092707 489 1097 2369 13 [3,-2,1,-1,1,-1,0,1,-1,0,0,1,-1]
** 1.000519113342433 377 988 1908 14 [3,-1,0,0,0,1,-2,0,0,0,0,-1,2,-1]
** 1.000460258220997 190 430 1862 21 [3,-1,-2,1,0,1,-1,-1,2,-1,-1,1,0,0,1,-1,0,0,0,1,-1]
** 1.000460258220997 430 973 1861 21 [3,-1,-2,1,0,1,-1,-1,2,-1,-1,1,0,0,1,-1,0,0,0,1,-1]
** 1.001009258596126 247 823 1795 16 [4,-3,4,-6,5,-4,5,-3,0,-1,-1,2,-1,1,-2,1]
** 1.000302834273007 306 1630 1730 10 [5,2,-1,-2,-1,2,2,-3,-1,-2]
** 1.000555891890885 241 700 1589 12 [3,0,-1,1,-1,-1,0,1,0,0,1,-2]
** 1.001437520701664 547 1974 1500 11 [3,2,1,-1,0,-1,0,1,0,-2,-2]
** 1.000316954940482 5 1977 1486 10 [395,158,0,-79,-79,-79,-79,-79,-79,-79]
** 1.000321024961523 5 1952 1467 10 [390,156,0,-78,-78,-78,-78,-78,-78,-78]
** 1.001597047141210 378 1501 1454 9 [4,-1,4,-2,1,-3,-1,0,-1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=1/5, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000035446251236 193 1893 10891 8 [11,-12,3,0,5,-5,3,-3]
** 1.000226247803807 372 832 8889 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1]
** 1.000226247803807 832 1861 8888 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1]
** 1.000101629707544 399 1831 8730 14 [4,2,3,1,0,0,0,-1,-2,-2,-1,-1,-1,-1]
** 1.000115254239615 114 825 8515 10 [7,2,-2,-1,2,1,1,0,1,-4]
** 1.000118136558231 207 1379 6740 6 [6,5,-4,0,2,-2]
** 1.000033867040633 79 1146 6661 9 [15,-7,-2,-6,4,6,-2,3,-9]
** 1.000146782468316 416 1264 6490 12 [3,0,1,-2,0,0,1,0,-2,-1,0,1]
** 1.000167938722114 297 1543 5979 12 [4,5,6,2,-2,-4,-5,-3,-3,0,1,1]
** 1.000194176476517 114 1726 5243 18 [15,2,2,-2,-2,0,2,0,-2,0,2,0,-2,0,2,0,-2,-2]
** 1.000152037699776 121 761 4790 6 [6,2,-1,-1,1,-4]
** 1.000193180960489 240 1510 4489 11 [7,-5,3,3,-3,2,-2,1,-1,1,-1]
** 1.000405435601516 269 744 4076 9 [3,0,-2,0,2,-1,-1,1,-1]
** 1.000189073335693 116 1504 3820 5 [12,12,7,-4,1]
** 1.000161465182546 431 1932 3801 8 [5,-3,4,-5,4,-4,3,-1]
** 1.000108265319972 103 1167 3647 4 [11,4,-2,-11]
** 1.000166937059998 263 1553 3456 13 [6,0,-3,-2,0,3,1,1,-2,-3,1,2,-2]
** 1.000279103859397 82 444 3145 8 [6,-3,-1,0,2,1,-2,-1]
** 1.000282757088752 110 495 2920 11 [4,3,-3,-2,1,0,-2,0,-1,0,2]
** 1.000463969727674 124 424 2662 7 [4,-3,4,-2,1,-1,-1]
** 1.000463969727674 424 1450 2661 7 [4,-3,4,-2,1,-1,-1]
** 1.000555891890885 241 700 2476 12 [3,0,-1,1,-1,-1,0,1,0,0,1,-2]
** 1.000467656763371 66 291 2291 14 [4,2,0,-4,0,2,2,-2,-2,0,4,0,-2,-3]
** 1.000467656763371 291 1283 2290 14 [4,2,0,-4,0,2,2,-2,-2,0,4,0,-2,-3]
** 1.000020550093545 303 947 2277 9 [3,0,1,1,-1,0,0,-1,-1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=1/4, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000089148691747 29 1070 26527 25 [37,-4,6,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,13,-10,-8,14]
** 1.000090412637074 29 1911 26137 25 [66,-7,11,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,23,-18,-14,25]
** 1.000035446251236 193 1893 17236 8 [11,-12,3,0,5,-5,3,-3]
** 1.000115254239615 114 825 10438 10 [7,2,-2,-1,2,1,1,0,1,-4]
** 1.000107273937310 386 1684 9345 8 [4,2,-2,1,-1,0,1,-2]
** 1.000226247803807 372 832 8827 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1]
** 1.000226247803807 832 1861 8826 15 [2,1,-1,0,0,-1,1,0,-1,1,0,-1,1,0,-1]
** 1.000118136558231 207 1379 8660 6 [6,5,-4,0,2,-2]
** 1.000034775162274 113 729 7924 11 [7,-4,3,0,-1,-3,-1,1,-3,1,2]
** 1.000116397870165 351 1097 6691 15 [3,1,-2,0,1,0,-1,0,2,0,-2,1,1,-1,-1]
** 1.000152037699776 121 761 6247 6 [6,2,-1,-1,1,-4]
** 1.000166040411401 224 1358 6086 25 [6,0,2,2,-1,-1,1,0,-1,0,0,-1,-1,0,0,-1,0,0,-1,0,0,-1,0,0,-1]
** 1.000108265319972 103 1167 5704 4 [11,4,-2,-11]
** 1.000146782468316 416 1264 5616 12 [3,0,1,-2,0,0,1,0,-2,-1,0,1]
** 1.000107700453608 156 1108 5377 8 [8,-7,5,-5,7,-7,2,1]
** 1.000161465182546 431 1932 5177 8 [5,-3,4,-5,4,-4,3,-1]
** 1.000166937059998 263 1553 4766 13 [6,0,-3,-2,0,3,1,1,-2,-3,1,2,-2]
** 1.000188345303973 49 1889 4066 13 [39,-17,-12,7,3,6,-19,5,-11,-1,8,-5,11]
** 1.000279103859397 82 444 3930 8 [6,-3,-1,0,2,1,-2,-1]
** 1.000282757088752 110 495 3716 11 [4,3,-3,-2,1,0,-2,0,-1,0,2]
** 1.000214865252435 88 1816 3261 4 [21,-8,11,-19]
** 1.000186647603218 32 1943 3075 12 [61,-17,-6,19,-2,6,-19,2,-6,17,6,-19]
** 1.000243869198878 246 1921 2937 9 [8,-2,4,0,-2,0,0,0,-2]
** 1.000502364766446 81 1973 2904 11 [25,-16,9,-7,2,3,-15,14,-4,-5,-3]
** 1.000198083371338 32 919 2888 12 [29,-8,-3,9,-1,3,-9,1,-3,8,3,-9]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=1/3, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000016313761905 38 1244 17476 6 [34,-43,55,-53,43,-25]
** 1.000042112147912 51 974 16081 9 [20,-18,15,-8,10,-5,-5,10,-12]
** 1.000064117179818 339 1874 11451 7 [7,-9,5,-2,5,-7,3]
** 1.000162039232093 436 1867 9563 8 [5,-3,-1,3,0,-1,-1,1]
** 1.000020717712129 30 1889 9201 4 [63,-3,57,-21]
** 1.000108265319972 103 1167 8318 4 [11,4,-2,-11]
** 1.000152037699776 121 761 8134 6 [6,2,-1,-1,1,-4]
** 1.000268505640947 732 1979 7537 11 [4,-4,1,1,0,-1,1,0,0,-1,1]
** 1.000186457220578 192 1720 6950 5 [9,0,-3,-4,7]
** 1.000036027319351 117 1416 6838 9 [12,1,3,-1,-3,1,3,-1,-3]
** 1.000079918001269 64 374 6478 14 [5,5,0,-3,1,3,-2,-1,2,1,-2,-4,-2,-1]
** 1.000156019281739 191 1211 6314 16 [6,3,-5,-2,-1,-1,-3,0,3,1,1,1,3,0,-3,-1]
** 1.000081935557988 488 1724 6051 10 [3,1,3,0,2,-2,0,-1,-1,-1]
** 1.000329313909862 213 502 6019 17 [3,-2,1,0,1,-1,1,0,1,-1,0,0,0,0,-1,1,-1]
** 1.000329313909862 502 1183 6018 17 [3,-2,1,0,1,-1,1,0,1,-1,0,0,0,0,-1,1,-1]
** 1.000279811372081 229 1413 5571 17 [6,1,0,2,-1,0,-2,0,0,1,0,0,-1,-1,0,0,1]
** 1.000371009548000 575 1430 5287 18 [2,1,1,-1,0,-1,0,-1,0,0,1,0,0,0,1,0,-1,-1]
** 1.000037597110473 30 989 5080 4 [33,-2,30,-11]
** 1.000200789673608 46 538 5003 8 [11,8,2,-5,-7,1,6,1]
** 1.000107364658279 121 943 4930 8 [7,6,1,3,2,-2,-4,-4]
** 1.000242412381303 537 1766 4728 9 [3,0,3,0,2,-2,0,-2,-1]
** 1.000214865252435 88 1816 4560 4 [21,-8,11,-19]
** 1.000185957868797 392 1785 4550 15 [5,-2,0,-1,1,2,0,-1,0,0,-1,0,1,0,-1]
** 1.000454297968062 494 1754 4497 14 [4,-2,2,-2,0,0,0,0,1,0,0,0,0,-1]
** 1.000137217936991 187 1366 4360 5 [7,2,1,4,5]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=2/5, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000032097044243 151 1862 34920 14 [12,4,1,0,-3,0,-3,0,-3,0,-3,-1,8,4]
** 1.000012412932087 41 316 32793 6 [8,-2,-2,-1,3,3]
** 1.000021753465840 76 819 27738 8 [10,8,4,-1,7,8,7,1]
** 1.000084064416757 788 1849 21122 18 [1,2,2,1,1,1,1,0,0,-1,0,0,0,-1,-1,-1,-1,-1]
** 1.000042112147912 51 974 20350 9 [20,-18,15,-8,10,-5,-5,10,-12]
** 1.000020717712129 30 1889 18023 4 [63,-3,57,-21]
** 1.000106871989807 145 1186 15541 7 [8,2,-5,5,3,-1,4]
** 1.000086884923348 62 970 13838 8 [15,10,2,-7,-9,2,8,1]
** 1.000039817648478 156 1734 13331 11 [12,-10,2,-2,4,0,-1,3,-8,9,-1]
** 1.000108548417854 151 1742 10818 4 [12,-5,-5,11]
** 1.000108265319972 103 1167 10000 4 [11,4,-2,-11]
** 1.000037597110473 30 989 9927 4 [33,-2,30,-11]
** 1.000102800010194 309 1835 9387 12 [5,5,3,2,2,2,-2,-3,-1,-1,-3,-4]
** 1.000081935557988 488 1724 8223 10 [3,1,3,0,2,-2,0,-1,-1,-1]
** 1.000186457220578 192 1720 7940 5 [9,0,-3,-4,7]
** 1.000096620275851 277 1909 7935 7 [8,-8,3,-4,4,-6,5]
** 1.000107364658279 121 943 6577 8 [7,6,1,3,2,-2,-4,-4]
** 1.000182159589929 155 863 6569 17 [6,-2,-2,-2,3,0,-2,1,1,-1,0,1,-1,0,1,0,-1]
** 1.000144591346480 37 998 6049 9 [28,-29,36,-28,29,-36,28,-28,8]
** 1.000200789673608 46 538 5934 8 [11,8,2,-5,-7,1,6,1]
** 1.000198409224041 349 1376 5857 8 [4,0,-1,0,2,-1,-2,2]
** 1.000137217936991 187 1366 5705 5 [7,2,1,4,5]
** 1.000284853321768 496 1684 5576 11 [3,1,1,1,-1,-2,0,0,1,0,1]
** 1.000185957868797 392 1785 5572 15 [5,-2,0,-1,1,2,0,-1,0,0,-1,0,1,0,-1]
** 1.000242412381303 537 1766 5477 9 [3,0,3,0,2,-2,0,-2,-1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=1/2, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000002113050868 152 1422 547614 7 [10,-6,0,-3,1,0,4]
** 1.000011829173147 100 1713 84147 6 [18,-15,2,-6,18,-6]
** 1.000013436208326 75 834 64069 10 [12,-10,2,4,2,-4,-1,10,-6,4]
** 1.000012943548178 141 1506 60351 9 [10,7,3,-1,-3,-1,7,4,5]
** 1.000054399161024 244 1460 40035 12 [6,0,-1,3,-3,-1,-4,-2,2,-3,1,4]
** 1.000023097532218 59 954 38697 10 [15,17,29,30,32,27,15,10,2,3]
** 1.000032904252395 38 1892 38619 13 [50,-11,26,10,23,30,6,27,26,10,25,-6,38]
** 1.000021753465840 76 819 37929 8 [10,8,4,-1,7,8,7,1]
** 1.000009119634599 72 1620 35338 5 [23,-11,-6,8,10]
** 1.000012766236563 104 1076 35337 9 [11,-7,2,4,1,-2,-1,1,1]
** 1.000065262933536 387 1579 32975 8 [4,0,1,1,1,1,2,1]
** 1.000020717712129 30 1889 28774 4 [63,-3,57,-21]
** 1.000057021897901 114 645 25959 13 [6,-2,1,-4,2,-1,1,-2,-2,2,0,3,-1]
** 1.000020673026896 169 1862 19088 6 [10,10,13,4,1,-6]
** 1.000039817648478 156 1734 18892 11 [12,-10,2,-2,4,0,-1,3,-8,9,-1]
** 1.000094740320442 150 1862 16454 7 [13,-8,10,-14,10,-8,12]
** 1.000034765624485 205 1299 16298 8 [6,2,1,-1,-1,4,2,3]
** 1.000037597110473 30 989 15889 4 [33,-2,30,-11]
** 1.000062871354546 155 1910 15450 9 [12,4,0,-3,-11,-1,4,5,7]
** 1.000067964806049 172 601 14748 16 [3,1,2,1,3,0,1,1,1,0,0,0,0,1,0,1]
** 1.000109013472929 127 1454 14657 7 [12,-7,9,-13,9,-7,11]
** 1.000108548417854 151 1742 13088 4 [12,-5,-5,11]
** 1.000108265319972 103 1167 12086 4 [11,4,-2,-11]
** 1.000058631039483 419 1971 12052 11 [5,-2,3,-1,2,-2,1,2,0,0,1]
** 1.000083374142554 256 1344 11461 8 [6,-5,6,-2,-2,5,-3,4]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=3/5, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000010533055300 112 1220 99210 9 [11,-2,9,0,5,2,8,-2,7]
** 1.000025958814849 126 1671 51989 12 [13,4,-7,1,-3,-6,-4,-3,3,6,3,-2]
** 1.000006955568155 126 1436 46928 9 [12,-7,1,5,1,-3,-1,1,1]
** 1.000012943548178 141 1506 43137 9 [10,7,3,-1,-3,-1,7,4,5]
** 1.000021753465840 76 819 27655 8 [10,8,4,-1,7,8,7,1]
** 1.000055662303369 53 747 26579 16 [15,-13,3,6,6,-10,6,1,9,-11,5,4,4,-8,8,-4]
** 1.000038363444793 108 886 26008 7 [9,-6,-5,6,-4,-4,6]
** 1.000053944430277 116 1336 23973 6 [12,-6,6,-11,2,1]
** 1.000020717712129 30 1889 18054 4 [63,-3,57,-21]
** 1.000086366805472 73 660 11943 12 [9,0,3,3,1,-1,-2,-1,0,1,1,1]
** 1.000117140587403 121 1323 11934 15 [11,-1,3,1,-2,0,2,0,-1,0,1,0,-1,0,1]
** 1.000060984318025 106 345 11659 11 [3,1,-1,1,2,-2,1,2,0,0,2]
** 1.000060984318025 345 1123 11658 11 [3,1,-1,1,2,-2,1,2,0,0,2]
** 1.000067964806049 172 601 11481 16 [3,1,2,1,3,0,1,1,1,0,0,0,0,1,0,1]
** 1.000009119634599 72 1620 10860 5 [23,-11,-6,8,10]
** 1.000053242638087 326 1922 10239 8 [7,-7,3,-1,3,-6,7,-3]
** 1.000108265319972 103 1167 10040 4 [11,4,-2,-11]
** 1.000037597110473 30 989 9958 4 [33,-2,30,-11]
** 1.000034765624485 205 1299 9887 8 [6,2,1,-1,-1,4,2,3]
** 1.000166574535744 304 952 9526 20 [3,1,-2,0,2,-1,-1,-1,1,0,-1,1,1,-1,-1,1,0,-1,0,1]
** 1.000090095647115 241 1661 9303 14 [7,-1,2,-2,3,0,-2,-4,2,-2,2,-2,3,1]
** 1.000133114927012 228 532 9201 16 [2,1,0,-1,-1,1,0,1,0,0,-1,-1,1,0,0,-1]
** 1.000133114927012 532 1241 9200 16 [2,1,0,-1,-1,1,0,1,0,0,-1,-1,1,0,0,-1]
** 1.000186457220578 192 1720 7944 5 [9,0,-3,-4,7]
** 1.000062941732713 517 1769 7371 9 [4,-2,0,0,1,-1,2,-1,1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=2/3, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000003542706183 142 1924 107470 9 [14,-6,-2,8,-4,5,1,-3,-2]
** 1.000047313152607 334 1398 21282 11 [5,-4,3,-2,-1,1,1,0,1,0,-1]
** 1.000102090094147 100 992 13290 15 [10,-1,2,1,-2,0,2,0,-1,0,1,0,-1,0,1]
** 1.000070737583946 41 469 10628 10 [11,5,1,-8,-1,1,11,-1,-4,-5]
** 1.000117140587403 121 1323 10396 15 [11,-1,3,1,-2,0,2,0,-1,0,1,0,-1,0,1]
** 1.000020717712129 30 1889 9243 4 [63,-3,57,-21]
** 1.000241946612386 22 98 9190 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1]
** 1.000241946612386 98 437 9189 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1]
** 1.000241946612386 437 1949 9188 25 [5,-3,3,-2,2,-1,1,0,0,1,-1,2,-1,-2,1,-1,1,-1,1,-1,1,-1,1,-1,1]
** 1.000166574535744 304 952 8457 20 [3,1,-2,0,2,-1,-1,-1,1,0,-1,1,1,-1,-1,1,0,-1,0,1]
** 1.000108265319972 103 1167 8358 4 [11,4,-2,-11]
** 1.000090095647115 241 1661 7293 14 [7,-1,2,-2,3,0,-2,-4,2,-2,2,-2,3,1]
** 1.000142443422050 59 1859 6791 15 [32,-16,16,1,2,4,9,2,5,-6,4,9,-14,4,-8]
** 1.000246850262805 454 1568 5927 10 [2,5,1,-3,-1,1,1,0,-2,-2]
** 1.000065916929107 48 1526 5787 11 [32,-6,-20,4,0,-21,14,18,-3,-10,2]
** 1.000132652778331 289 1837 5749 6 [6,2,1,4,3,-2]
** 1.000270955335396 402 1717 5706 13 [4,1,1,-2,3,-1,0,-2,1,-2,-2,0,1]
** 1.000166590432859 115 1758 5702 13 [15,4,6,0,-4,0,4,2,-2,-2,0,2,2]
** 1.000190145788484 110 686 5219 5 [7,-4,-5,1,3]
** 1.000048962251457 307 1508 5126 7 [5,0,-2,-1,2,2,-4]
** 1.000037597110473 30 989 5111 4 [33,-2,30,-11]
** 1.000346620339653 68 1366 5019 8 [19,21,17,9,-2,-6,-15,-13]
** 1.000234436716968 170 1062 4818 8 [6,2,-3,1,1,-4,1,5]
** 1.000081284932752 50 295 4676 9 [6,-1,3,-3,0,-2,-1,2,-2]
** 1.000081284932752 295 1741 4675 9 [6,-1,3,-3,0,-2,-1,2,-2]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=3/4, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000118862375490 267 1605 11953 7 [5,6,1,-3,-1,0,-2]
** 1.000098076956125 235 563 5705 11 [2,1,0,0,-1,1,0,-1,-1,0,1]
** 1.000098076956125 563 1349 5704 11 [2,1,0,0,-1,1,0,-1,-1,0,1]
** 1.000108265319972 103 1167 5664 4 [11,4,-2,-11]
** 1.000270955335396 402 1717 4646 13 [4,1,1,-2,3,-1,0,-2,1,-2,-2,0,1]
** 1.000093030795381 136 440 4085 8 [3,1,-1,1,0,-2,-1,1]
** 1.000093030795381 440 1424 4084 8 [3,1,-1,1,0,-2,-1,1]
** 1.000305043968183 134 1623 3803 16 [13,-11,3,0,-2,-2,-1,1,1,1,0,-1,-1,0,1,1]
** 1.000202028544894 81 542 3728 11 [7,-2,0,-3,2,-1,-1,-4,1,2,1]
** 1.000132652778331 289 1837 3569 6 [6,2,1,4,3,-2]
** 1.000314732063239 36 1085 3554 13 [30,4,6,-4,-5,-1,5,1,-5,-1,5,1,-5]
** 1.000616540428395 320 1279 3202 17 [4,0,0,0,-1,1,-1,1,-1,1,-1,1,0,-3,0,0,1]
** 1.000962235223722 315 768 2857 10 [3,-2,2,-1,0,-1,1,-1,1,-1]
** 1.000962235223722 768 1873 2856 10 [3,-2,2,-1,0,-1,1,-1,1,-1]
** 1.000815453738719 97 1052 2458 17 [10,9,2,-1,0,0,1,4,-2,-1,-3,-7,-4,0,-2,-4,2]
** 1.000659710741214 96 684 2435 8 [7,1,0,-5,-2,0,4,-2]
** 1.000539486988851 310 814 2409 9 [2,2,-1,0,1,-1,-1,0,1]
** 1.000642177222389 182 609 2375 9 [4,-3,3,-1,0,1,-1,1,-2]
** 1.000121054039365 22 93 2324 11 [4,1,0,-1,1,0,0,-3,0,-1,1]
** 1.000121054039365 93 393 2323 11 [4,1,0,-1,1,0,0,-3,0,-1,1]
** 1.000121054039365 393 1661 2322 11 [4,1,0,-1,1,0,0,-3,0,-1,1]
** 1.000550715091592 112 1830 2229 6 [16,5,9,-2,7,-7]
** 1.000464938518246 133 1057 2219 10 [9,-9,6,-8,6,-7,5,-2,6,-4]
** 1.000379615738119 64 1457 2174 4 [23,-6,16,-16]
** 1.000701863506824 247 966 2049 21 [5,-5,3,0,-2,2,-1,0,1,-1,1,-1,0,1,-1,1,0,-1,1,-1,1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=4/5, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000230561690733 587 1655 9512 11 [2,2,1,0,0,-2,-2,-1,0,1,1]
** 1.000216742303556 93 1010 3622 15 [11,-2,5,2,9,-3,5,-2,-2,-2,-3,-5,-4,1,-7]
** 1.000588628388860 205 439 3344 20 [3,-2,1,-2,1,1,-1,1,-2,2,-2,2,-1,1,-1,-1,2,-1,1,-1]
** 1.000588628388860 439 940 3343 20 [3,-2,1,-2,1,1,-1,1,-2,2,-2,2,-1,1,-1,-1,2,-1,1,-1]
** 1.000498435859590 580 1483 2852 15 [3,-1,0,-1,0,1,0,-1,0,1,0,0,-1,-1,1]
** 1.000755504545288 372 1126 2703 15 [3,0,1,-2,-1,0,2,0,0,-2,-1,1,0,1,-1]
** 1.000202028544894 81 542 2595 11 [7,-2,0,-3,2,-1,-1,-4,1,2,1]
** 1.000465887867211 416 1393 2399 13 [3,1,1,-1,-1,-2,1,-1,1,-1,0,0,1]
** 1.000369622285660 652 1547 2335 17 [3,-1,-1,-1,1,2,-2,0,0,0,1,-1,0,0,0,1,-1]
** 1.000727451123105 150 1721 2267 15 [11,5,5,0,-3,-4,-12,0,-1,1,-1,0,1,-1,1]
** 1.000659710741214 96 684 2077 8 [7,1,0,-5,-2,0,4,-2]
** 1.000607401714680 326 856 2040 10 [3,-1,0,0,1,-2,1,-1,1,-1]
** 1.000177346113274 44 1648 2024 10 [39,-59,42,-10,-9,19,-10,-9,20,-17]
** 1.000390540272179 610 1596 1999 15 [3,-1,0,0,0,0,0,-1,0,0,0,0,0,1,-1]
** 1.000642177222389 182 609 1992 9 [4,-3,3,-1,0,1,-1,1,-2]
** 1.000832153882265 297 1732 1910 19 [6,-1,0,1,-2,1,0,1,-2,1,1,-1,0,-4,0,0,-1,2,-1]
** 1.000132652778331 289 1837 1891 6 [6,2,1,4,3,-2]
** 1.000550715091592 112 1830 1812 6 [16,5,9,-2,7,-7]
** 1.000659981878357 266 1084 1766 6 [5,-4,1,-1,4,-3]
** 1.000968314086598 200 614 1659 14 [3,0,1,-1,0,-1,2,0,0,-1,-1,-1,1,-1]
** 1.000968314086598 614 1885 1658 14 [3,0,1,-1,0,-1,2,0,0,-1,-1,-1,1,-1]
** 1.000379615738119 64 1457 1628 4 [23,-6,16,-16]
** 1.000734414794903 482 1418 1493 8 [3,-1,3,-2,2,-3,1,-1]
** 1.000134179125879 91 914 1484 13 [11,-10,4,0,-2,-2,-1,1,1,1,0,0,-1]
** 1.000699186097681 105 396 1460 15 [3,3,0,-1,-1,1,1,-1,-1,1,1,0,-1,-3,-1]
** ----------------- ---- ---- ------ -- ------------
**
** 25 longest fake sequences for r=1, 2 <= x <= 2000, x+2 <= y <= 2000
** root x y fail o coefficients
** ----------------- ---- ---- ------ -- ------------
** 1.000028402156455 450 1695 38415 14 [4,-1,1,-2,0,-1,3,-2,0,0,0,0,1,-2]
** 1.000190482197144 124 390 13414 13 [4,-3,1,0,0,0,-1,1,0,-1,0,2,-2]
** 1.000190482197144 390 1227 13413 13 [4,-3,1,0,0,0,-1,1,0,-1,0,2,-2]
** 1.000143584240061 275 1330 6504 10 [5,-1,1,0,1,-2,-1,-1,-2,1]
** 1.000437651448089 597 1940 4341 10 [4,-3,2,0,-2,1,-1,-1,2,-1]
** 1.000544776855336 25 84 4099 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2]
** 1.000544776855336 84 283 4098 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2]
** 1.000544776855336 283 954 4097 12 [4,-2,0,-2,2,1,-1,-2,1,1,1,-2]
** 1.000304050189770 324 1304 3739 9 [3,4,1,-1,-3,-3,-2,0,2]
** 1.000257081176362 92 548 3379 8 [5,5,4,2,-3,-6,-4,-2]
** 1.000438783587243 605 1577 2856 12 [3,-1,0,0,-1,2,-1,-1,0,0,1,-1]
** 1.000655507570736 176 587 2401 8 [3,1,1,-1,-3,-1,0,1]
** 1.000655507570736 587 1958 2400 8 [3,1,1,-1,-3,-1,0,1]
** 1.000890787851308 374 1506 2387 9 [4,0,1,-2,-1,0,1,-1,-1]
** 1.000689380300664 81 269 2261 12 [4,-2,-1,1,-2,3,-2,0,1,-1,2,-2]
** 1.000689380300664 269 894 2260 12 [4,-2,-1,1,-2,3,-2,0,1,-1,2,-2]
** 1.000196961275789 642 1965 2232 12 [3,1,-2,-2,1,2,0,-3,0,1,1,-1]
** 1.000930815480001 420 1766 2004 14 [5,-4,3,-1,0,0,1,-1,-1,2,-4,3,-3,1]
** 1.000916107354094 319 1505 1815 10 [5,-1,-2,2,1,-2,-1,2,0,-3]
** 1.000605568483049 170 1790 1802 11 [10,5,6,1,-2,0,3,-3,-4,-8,-7]
** 1.000817823206005 206 495 1756 10 [3,-1,-1,0,-1,2,0,-1,1,-1]
** 1.000817823206005 495 1190 1755 10 [3,-1,-1,0,-1,2,0,-1,1,-1]
** 1.001326318073690 458 1409 1617 10 [3,1,-2,-1,0,-1,0,3,0,-2]
** 1.000499490255901 165 425 1613 12 [3,-1,0,-1,1,0,1,-2,1,-1,1,-1]
** 1.000499490255901 425 1095 1612 12 [3,-1,0,-1,1,0,1,-2,1,-1,1,-1]
** ----------------- ---- ---- ------ -- ------------
**
*/