# Extensive Pisot Data by Tomas Oliveira e Silva

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]
**   ----------------- ---- ---- ------ -- ------------
**
*/