A list of Succesful Recurrences or Order<=, 6, for the Infinite Families of Pisot Sequence with parameter, 1/2 with initial conditions x, x^2*k+y, for x from 2 to , 20, and y from 1 to x^2-1 that is not a multiple of x For GENERAL (symbolic!) k By Shalosh B. Ekhad The Pisot Sequence a(n)=PS(x,y)(n), is defined by, a(1) = x, a(2) = y, 2 a(n + 1) and for n>1, a(n + 2) = trunc(1/2 + ---------) a(n) In this gripping article, we will list PROVED recurrences for the Pisot Seq\ uenes 2 PISOT(x, k x + y) for all (numeric) integers x between 2 and, 20, and all numeric integers, y, that are not divisible by x from 1 to x^2-1. We will use the convention that a linear recurrence of order r a[n]=C1*a[n-1]+ ... +Cr*a[n-r] is abbreviated as a pair of lists: [[a[1], ..., a[r]],[C1, ..., Cr]], where \ the first list consists of the r initial values . PISOT(2, 4 k + 1) = [[2, 4 k + 1], [1 + 2 k, -k]] PISOT(2, 4 k + 3) = [[2, 4 k + 3], [1 + 2 k, 1 + k]] PISOT(3, 9 k + 1) = [[3, 9 k + 1], [3 k, k]] 2 PISOT(3, 9 k + 2) = [[3, 9 k + 2, 27 k + 12 k + 1], [3 k, 2 k, k]] PISOT(3, 9 k + 4) = [[3, 9 k + 4], [2 + 3 k, -1 - 2 k]] PISOT(3, 9 k + 5) = [[3, 9 k + 5], [1 + 3 k, 1 + 2 k]] PISOT(3, 9 k + 7) = 2 [[3, 9 k + 7, 27 k + 42 k + 16], [3 + 3 k, -2 - 2 k, 1 + k]] PISOT(3, 9 k + 8) = [[3, 9 k + 8], [3 + 3 k, -1 - k]] PISOT(4, 16 k + 1) = [[4, 16 k + 1], [4 k, k]] 2 PISOT(4, 16 k + 2) = [[4, 16 k + 2, 64 k + 16 k + 1], [1 + 4 k, -2 k, -k]] PISOT(4, 16 k + 5) = [[4, 16 k + 5], [2 + 4 k, -1 - 3 k]] PISOT(4, 16 k + 7) = 2 [[4, 16 k + 7, 64 k + 56 k + 12], [2 + 4 k, -1 - k, 1 + 2 k]] 2 PISOT(4, 16 k + 9) = [[4, 16 k + 9, 64 k + 72 k + 20], [2 + 4 k, k, 1 + 2 k]] PISOT(4, 16 k + 10) = 2 [[4, 16 k + 10, 64 k + 80 k + 25], [3 + 4 k, -2 - 2 k, 2 + 3 k]] PISOT(4, 16 k + 11) = [[4, 16 k + 11], [2 + 4 k, 2 + 3 k]] PISOT(4, 16 k + 14) = 2 [[4, 16 k + 14, 64 k + 112 k + 49], [4 + 4 k, -2 - 2 k, 1 + k]] PISOT(4, 16 k + 15) = [[4, 16 k + 15], [4 + 4 k, -1 - k]] PISOT(5, 25 k + 1) = [[5, 25 k + 1], [5 k, k]] 2 PISOT(5, 25 k + 2) = [[5, 25 k + 2, 125 k + 20 k + 1], [5 k, 2 k, k]] PISOT(5, 25 k + 3) = [ 2 3 2 [5, 25 k + 3, 125 k + 30 k + 2, 625 k + 225 k + 29 k + 1], [5 k, 3 k, 2 k, k]] PISOT(5, 25 k + 6) = [[5, 25 k + 6], [2 + 5 k, -1 - 4 k]] PISOT(5, 25 k + 8) = [[5, 25 k + 8], [1 + 5 k, 1 + 3 k]] PISOT(5, 25 k + 9) = 2 [[5, 25 k + 9, 125 k + 90 k + 16], [2 + 5 k, -1 - k, 1 + 3 k]] PISOT(5, 25 k + 11) = [ 2 3 2 [5, 25 k + 11, 125 k + 110 k + 24, 625 k + 825 k + 361 k + 52], [3 + 5 k, -2 - 4 k, k, 1 + 2 k]] PISOT(5, 25 k + 12) = [[5, 25 k + 12], [2 + 5 k, 1 + 2 k]] PISOT(5, 25 k + 13) = [[5, 25 k + 13], [3 + 5 k, -1 - 2 k]] PISOT(5, 25 k + 14) = [ 2 3 2 [5, 25 k + 14, 125 k + 140 k + 39, 625 k + 1050 k + 586 k + 109], [2 + 5 k, 2 + 4 k, 1 + k, -1 - 2 k]] PISOT(5, 25 k + 16) = 2 [[5, 25 k + 16, 125 k + 160 k + 51], [3 + 5 k, k, 2 + 3 k]] PISOT(5, 25 k + 17) = [[5, 25 k + 17], [4 + 5 k, -2 - 3 k]] PISOT(5, 25 k + 19) = [[5, 25 k + 19], [3 + 5 k, 3 + 4 k]] PISOT(5, 25 k + 22) = [ 2 3 2 [5, 25 k + 22, 125 k + 220 k + 97, 625 k + 1650 k + 1454 k + 428], [5 + 5 k, -3 - 3 k, 2 + 2 k, -1 - k]] PISOT(5, 25 k + 23) = 2 [[5, 25 k + 23, 125 k + 230 k + 106], [5 + 5 k, -2 - 2 k, 1 + k]] PISOT(5, 25 k + 24) = [[5, 25 k + 24], [5 + 5 k, -1 - k]] PISOT(6, 36 k + 1) = [[6, 36 k + 1], [6 k, k]] 2 PISOT(6, 36 k + 2) = [[6, 36 k + 2, 216 k + 24 k + 1], [6 k, 2 k, k]] PISOT(6, 36 k + 3) = [ 2 3 2 [6, 36 k + 3, 216 k + 36 k + 2, 1296 k + 324 k + 33 k + 1], [1 + 6 k, -3 k, -k, -k]] 2 PISOT(6, 36 k + 4) = [[6, 36 k + 4, 216 k + 48 k + 3, 3 2 4 3 2 1296 k + 432 k + 52 k + 2, 7776 k + 3456 k + 612 k + 48 k + 1], [6 k, 4 k, 3 k, 2 k, k]] PISOT(6, 36 k + 7) = [[6, 36 k + 7], [2 + 6 k, -1 - 5 k]] PISOT(6, 36 k + 8) = [ 2 3 2 [6, 36 k + 8, 216 k + 96 k + 11, 1296 k + 864 k + 196 k + 15], [1 + 6 k, 1 + 2 k, -3 k, -1 - 4 k]] 2 PISOT(6, 36 k + 11) = [[6, 36 k + 11, 216 k + 132 k + 20, 3 2 4 3 2 1296 k + 1188 k + 361 k + 36, 7776 k + 9504 k + 4338 k + 872 k + 65, 5 4 3 2 46656 k + 71280 k + 43416 k + 13139 k + 1972 k + 117], [2 + 6 k, -1 - k, 1 + 4 k, k, 2 k, 1 + 3 k]] PISOT(6, 36 k + 13) = 2 [[6, 36 k + 13, 216 k + 156 k + 28], [1 + 6 k, 2 + 7 k, 1 + 3 k]] PISOT(6, 36 k + 14) = 2 [[6, 36 k + 14, 216 k + 168 k + 33], [2 + 6 k, 2 k, 2 + 5 k]] 2 PISOT(6, 36 k + 15) = [[6, 36 k + 15, 216 k + 180 k + 38, 3 2 4 3 2 1296 k + 1620 k + 681 k + 96, 7776 k + 12960 k + 8154 k + 2292 k + 243 ], [3 + 6 k, -1 - 3 k, -k, -2 - 3 k, 2 + 5 k]] No uniform order PISOT(6, 36 k + 17) = 2 [[6, 36 k + 17, 216 k + 204 k + 48], [2 + 6 k, 2 + 5 k, 1 + 2 k]] PISOT(6, 36 k + 19) = 2 [[6, 36 k + 19, 216 k + 228 k + 60], [4 + 6 k, -3 - 5 k, 1 + 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.997933255553849 2 PISOT(6, 36 k + 20) = [[6, 36 k + 20, 216 k + 240 k + 67, 3 2 1296 k + 2160 k + 1204 k + 224, 4 3 2 7776 k + 17280 k + 14436 k + 5368 k + 749], [3 + 6 k, 1 + 2 k, 1 + k, -2 - 3 k, 1 + 2 k]] No uniform order PISOT(6, 36 k + 22) = 2 [[6, 36 k + 22, 216 k + 264 k + 81], [4 + 6 k, -2 - 2 k, 3 + 5 k]] PISOT(6, 36 k + 23) = 2 [[6, 36 k + 23, 216 k + 276 k + 88], [5 + 6 k, -5 - 7 k, 2 + 3 k]] 2 PISOT(6, 36 k + 25) = [[6, 36 k + 25, 216 k + 300 k + 104, 3 2 1296 k + 2700 k + 1873 k + 433, 4 3 2 7776 k + 21600 k + 22482 k + 10396 k + 1803, 5 4 3 2 46656 k + 162000 k + 224856 k + 155989 k + 54102 k + 7508], [4 + 6 k, k, 3 + 4 k, -1 - k, 2 + 2 k, -2 - 3 k]] PISOT(6, 36 k + 28) = [ 2 3 2 [6, 36 k + 28, 216 k + 336 k + 131, 1296 k + 3024 k + 2356 k + 613], [5 + 6 k, -1 - 2 k, -3 - 3 k, 3 + 4 k]] PISOT(6, 36 k + 29) = [[6, 36 k + 29], [4 + 6 k, 4 + 5 k]] 2 PISOT(6, 36 k + 32) = [[6, 36 k + 32, 216 k + 384 k + 171, 3 2 1296 k + 3456 k + 3076 k + 914, 4 3 2 7776 k + 27648 k + 36900 k + 21912 k + 4885], [6 + 6 k, -4 - 4 k, 3 + 3 k, -2 - 2 k, 1 + k]] PISOT(6, 36 k + 33) = [ 2 3 2 [6, 36 k + 33, 216 k + 396 k + 182, 1296 k + 3564 k + 3273 k + 1004], [5 + 6 k, 3 + 3 k, -1 - k, 1 + k]] PISOT(6, 36 k + 34) = 2 [[6, 36 k + 34, 216 k + 408 k + 193], [6 + 6 k, -2 - 2 k, 1 + k]] PISOT(6, 36 k + 35) = [[6, 36 k + 35], [6 + 6 k, -1 - k]] PISOT(7, 49 k + 1) = [[7, 49 k + 1], [7 k, k]] 2 PISOT(7, 49 k + 2) = [[7, 49 k + 2, 343 k + 28 k + 1], [7 k, 2 k, k]] 2 PISOT(7, 49 k + 3) = [[7, 49 k + 3, 343 k + 42 k + 1], [7 k, 3 k, k]] PISOT(7, 49 k + 4) = [ 2 3 2 [7, 49 k + 4, 343 k + 56 k + 2, 2401 k + 588 k + 44 k + 1], [1 + 7 k, -3 k, -2 k, -k]] PISOT(7, 49 k + 8) = [[7, 49 k + 8], [2 + 7 k, -1 - 6 k]] PISOT(7, 49 k + 9) = 2 [[7, 49 k + 9, 343 k + 126 k + 12], [7 k, 1 + 9 k, 1 + 5 k]] PISOT(7, 49 k + 12) = 2 [[7, 49 k + 12, 343 k + 168 k + 21], [2 + 7 k, -1 - 2 k, 1 + 4 k]] PISOT(7, 49 k + 13) = 2 [[7, 49 k + 13, 343 k + 182 k + 24], [1 + 7 k, 1 + 6 k, 1 + 4 k]] PISOT(7, 49 k + 15) = [ 2 3 2 [7, 49 k + 15, 343 k + 210 k + 32, 2401 k + 2205 k + 673 k + 68], [2 + 7 k, k, 1 + 2 k, -1 - 3 k]] PISOT(7, 49 k + 16) = 2 [[7, 49 k + 16, 343 k + 224 k + 37], [3 + 7 k, -2 - 5 k, 1 + 3 k]] PISOT(7, 49 k + 17) = [[7, 49 k + 17], [2 + 7 k, 1 + 3 k]] PISOT(7, 49 k + 18) = [ 2 3 2 [7, 49 k + 18, 343 k + 252 k + 46, 2401 k + 2646 k + 968 k + 118], [2 + 7 k, 2 + 4 k, -1 - 4 k, -1 - 3 k]] PISOT(7, 49 k + 19) = [[7, 49 k + 19], [2 + 7 k, 2 + 5 k]] PISOT(7, 49 k + 20) = 2 [[7, 49 k + 20, 343 k + 280 k + 57], [4 + 7 k, -4 - 8 k, 2 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.993167813742437 PISOT(7, 49 k + 22) = [ 2 3 2 [7, 49 k + 22, 343 k + 308 k + 69, 2401 k + 3234 k + 1450 k + 216], [2 + 7 k, 3 + 8 k, 2 + 4 k, -1 - 2 k]] PISOT(7, 49 k + 23) = [[7, 49 k + 23], [3 + 7 k, 1 + 2 k]] PISOT(7, 49 k + 24) = [[7, 49 k + 24], [4 + 7 k, -2 - 4 k]] PISOT(7, 49 k + 25) = [[7, 49 k + 25], [3 + 7 k, 2 + 4 k]] PISOT(7, 49 k + 26) = [[7, 49 k + 26], [4 + 7 k, -1 - 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.993159161676151 PISOT(7, 49 k + 27) = [ 2 3 2 [7, 49 k + 27, 343 k + 378 k + 104, 2401 k + 3969 k + 2185 k + 401], [5 + 7 k, -5 - 8 k, 2 + 4 k, 1 + 2 k]] PISOT(7, 49 k + 29) = 2 [[7, 49 k + 29, 343 k + 406 k + 120], [3 + 7 k, 4 + 8 k, 3 + 5 k]] PISOT(7, 49 k + 30) = [[7, 49 k + 30], [5 + 7 k, -3 - 5 k]] PISOT(7, 49 k + 31) = [ 2 3 2 [7, 49 k + 31, 343 k + 434 k + 137, 2401 k + 4557 k + 2879 k + 605], [5 + 7 k, -2 - 4 k, -3 - 4 k, 2 + 3 k]] PISOT(7, 49 k + 32) = [[7, 49 k + 32], [5 + 7 k, -2 - 3 k]] PISOT(7, 49 k + 33) = 2 [[7, 49 k + 33, 343 k + 462 k + 156], [4 + 7 k, 3 + 5 k, 2 + 3 k]] PISOT(7, 49 k + 34) = [ 2 3 2 [7, 49 k + 34, 343 k + 476 k + 165, 2401 k + 4998 k + 3466 k + 801], [5 + 7 k, -1 - k, 1 + 2 k, 2 + 3 k]] PISOT(7, 49 k + 36) = 2 [[7, 49 k + 36, 343 k + 504 k + 185], [6 + 7 k, -5 - 6 k, 3 + 4 k]] PISOT(7, 49 k + 37) = 2 [[7, 49 k + 37, 343 k + 518 k + 196], [5 + 7 k, 1 + 2 k, 3 + 4 k]] PISOT(7, 49 k + 40) = 2 [[7, 49 k + 40, 343 k + 560 k + 229], [7 + 7 k, -8 - 9 k, 4 + 5 k]] PISOT(7, 49 k + 41) = [[7, 49 k + 41], [5 + 7 k, 5 + 6 k]] PISOT(7, 49 k + 45) = [ 2 3 2 [7, 49 k + 45, 343 k + 630 k + 289, 2401 k + 6615 k + 6071 k + 1856], [6 + 7 k, 3 + 3 k, -2 - 2 k, 1 + k]] PISOT(7, 49 k + 46) = 2 [[7, 49 k + 46, 343 k + 644 k + 302], [7 + 7 k, -3 - 3 k, 1 + k]] PISOT(7, 49 k + 47) = 2 [[7, 49 k + 47, 343 k + 658 k + 316], [7 + 7 k, -2 - 2 k, 1 + k]] PISOT(7, 49 k + 48) = [[7, 49 k + 48], [7 + 7 k, -1 - k]] PISOT(8, 64 k + 1) = [[8, 64 k + 1], [8 k, k]] 2 PISOT(8, 64 k + 2) = [[8, 64 k + 2, 512 k + 32 k + 1], [8 k, 2 k, k]] 2 PISOT(8, 64 k + 3) = [[8, 64 k + 3, 512 k + 48 k + 1], [8 k, 3 k, k]] PISOT(8, 64 k + 4) = [ 2 3 2 [8, 64 k + 4, 512 k + 64 k + 2, 4096 k + 768 k + 48 k + 1], [1 + 8 k, -4 k, -2 k, -k]] 2 PISOT(8, 64 k + 5) = [[8, 64 k + 5, 512 k + 80 k + 3, 3 2 4 3 2 4096 k + 960 k + 73 k + 2, 32768 k + 10240 k + 1176 k + 62 k + 1], [8 k, 5 k, 3 k, 2 k, k]] PISOT(8, 64 k + 9) = [[8, 64 k + 9], [2 + 8 k, -1 - 7 k]] PISOT(8, 64 k + 11) = [ 2 3 2 [8, 64 k + 11, 512 k + 176 k + 15, 4096 k + 2112 k + 361 k + 20], [1 + 8 k, 1 + 3 k, -4 k, -1 - 6 k]] PISOT(8, 64 k + 13) = [[8, 64 k + 13], [1 + 8 k, 1 + 5 k]] PISOT(8, 64 k + 15) = [ 2 3 2 [8, 64 k + 15, 512 k + 240 k + 28, 4096 k + 2880 k + 673 k + 52], [2 + 8 k, -k, -1 - 2 k, 1 + 4 k]] PISOT(8, 64 k + 17) = [ 2 3 2 [8, 64 k + 17, 512 k + 272 k + 36, 4096 k + 3264 k + 865 k + 76], [2 + 8 k, k, 2 k, 1 + 4 k]] PISOT(8, 64 k + 18) = 2 [[8, 64 k + 18, 512 k + 288 k + 41], [1 + 8 k, 2 + 10 k, 2 + 7 k]] PISOT(8, 64 k + 20) = [ 2 3 2 [8, 64 k + 20, 512 k + 320 k + 50, 4096 k + 3840 k + 1200 k + 125], [3 + 8 k, -1 - 4 k, -1 - 2 k, 1 + 3 k]] PISOT(8, 64 k + 21) = [[8, 64 k + 21], [3 + 8 k, -1 - 3 k]] PISOT(8, 64 k + 22) = 2 [[8, 64 k + 22, 512 k + 352 k + 61], [3 + 8 k, -1 - 2 k, 1 + 3 k]] PISOT(8, 64 k + 23) = [ 2 3 2 [8, 64 k + 23, 512 k + 368 k + 66, 4096 k + 4416 k + 1585 k + 189], [2 + 8 k, 2 + 7 k, 1 + 4 k, 1 + 3 k]] PISOT(8, 64 k + 25) = 2 [[8, 64 k + 25, 512 k + 400 k + 78], [3 + 8 k, 1 + k, -2 - 5 k]] PISOT(8, 64 k + 26) = 2 [[8, 64 k + 26, 512 k + 416 k + 85], [4 + 8 k, -3 - 6 k, 2 + 5 k]] PISOT(8, 64 k + 27) = 2 [[8, 64 k + 27, 512 k + 432 k + 91], [4 + 8 k, -3 - 5 k, 3 + 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.992185566218502 PISOT(8, 64 k + 28) = [ 2 3 2 [8, 64 k + 28, 512 k + 448 k + 98, 4096 k + 5376 k + 2352 k + 343], [3 + 8 k, 2 + 4 k, -2 k, -3 - 7 k]] PISOT(8, 64 k + 31) = 2 [[8, 64 k + 31, 512 k + 496 k + 120], [4 + 8 k, -1 - k, 2 + 4 k]] PISOT(8, 64 k + 33) = 2 [[8, 64 k + 33, 512 k + 528 k + 136], [4 + 8 k, k, 2 + 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.992191523674274 PISOT(8, 64 k + 36) = [ 2 3 2 [8, 64 k + 36, 512 k + 576 k + 162, 4096 k + 6912 k + 3888 k + 729], [5 + 8 k, -2 - 4 k, -2 - 2 k, 4 + 7 k]] PISOT(8, 64 k + 37) = 2 [[8, 64 k + 37, 512 k + 592 k + 171], [4 + 8 k, 2 + 5 k, 4 + 7 k]] PISOT(8, 64 k + 38) = 2 [[8, 64 k + 38, 512 k + 608 k + 181], [4 + 8 k, 3 + 6 k, 3 + 5 k]] PISOT(8, 64 k + 39) = 2 [[8, 64 k + 39, 512 k + 624 k + 190], [5 + 8 k, -k, -3 - 5 k]] PISOT(8, 64 k + 41) = [ 2 3 2 [8, 64 k + 41, 512 k + 656 k + 210, 4096 k + 7872 k + 5041 k + 1076], [6 + 8 k, -5 - 7 k, 3 + 4 k, -2 - 3 k]] PISOT(8, 64 k + 42) = 2 [[8, 64 k + 42, 512 k + 672 k + 221], [5 + 8 k, 1 + 2 k, 2 + 3 k]] PISOT(8, 64 k + 43) = [[8, 64 k + 43], [5 + 8 k, 2 + 3 k]] PISOT(8, 64 k + 44) = [ 2 3 2 [8, 64 k + 44, 512 k + 704 k + 242, 4096 k + 8448 k + 5808 k + 1331], [5 + 8 k, 3 + 4 k, -1 - 2 k, -2 - 3 k]] PISOT(8, 64 k + 46) = 2 [[8, 64 k + 46, 512 k + 736 k + 265], [7 + 8 k, -8 - 10 k, 5 + 7 k]] PISOT(8, 64 k + 47) = [ 2 3 2 [8, 64 k + 47, 512 k + 752 k + 276, 4096 k + 9024 k + 6625 k + 1621], [6 + 8 k, -1 - k, 2 + 2 k, -3 - 4 k]] PISOT(8, 64 k + 49) = [ 2 3 2 [8, 64 k + 49, 512 k + 784 k + 300, 4096 k + 9408 k + 7201 k + 1837], [6 + 8 k, 1 + k, -1 - 2 k, -3 - 4 k]] PISOT(8, 64 k + 51) = [[8, 64 k + 51], [7 + 8 k, -4 - 5 k]] PISOT(8, 64 k + 53) = [ 2 3 2 [8, 64 k + 53, 512 k + 848 k + 351, 4096 k + 10176 k + 8425 k + 2325], [7 + 8 k, -2 - 3 k, -4 - 4 k, 5 + 6 k]] PISOT(8, 64 k + 55) = [[8, 64 k + 55], [6 + 8 k, 6 + 7 k]] 2 PISOT(8, 64 k + 59) = [[8, 64 k + 59, 512 k + 944 k + 435, 3 2 4096 k + 11328 k + 10441 k + 3207, 4 3 2 32768 k + 120832 k + 167064 k + 102642 k + 23643], [8 + 8 k, -5 - 5 k, 3 + 3 k, -2 - 2 k, 1 + k]] PISOT(8, 64 k + 60) = [ 2 3 2 [8, 64 k + 60, 512 k + 960 k + 450, 4096 k + 11520 k + 10800 k + 3375], [7 + 8 k, 4 + 4 k, -2 - 2 k, 1 + k]] PISOT(8, 64 k + 61) = 2 [[8, 64 k + 61, 512 k + 976 k + 465], [8 + 8 k, -3 - 3 k, 1 + k]] PISOT(8, 64 k + 62) = 2 [[8, 64 k + 62, 512 k + 992 k + 481], [8 + 8 k, -2 - 2 k, 1 + k]] PISOT(8, 64 k + 63) = [[8, 64 k + 63], [8 + 8 k, -1 - k]] PISOT(9, 81 k + 1) = [[9, 81 k + 1], [9 k, k]] PISOT(9, 81 k + 2) = [[9, 81 k + 2], [9 k, 2 k]] 2 PISOT(9, 81 k + 3) = [[9, 81 k + 3, 729 k + 54 k + 1], [9 k, 3 k, k]] PISOT(9, 81 k + 4) = [ 2 3 2 [9, 81 k + 4, 729 k + 72 k + 2, 6561 k + 972 k + 52 k + 1], [9 k, 4 k, 2 k, k]] 2 PISOT(9, 81 k + 5) = [[9, 81 k + 5, 729 k + 90 k + 3, 3 2 4 3 2 6561 k + 1215 k + 79 k + 2, 59049 k + 14580 k + 1404 k + 66 k + 1], [1 + 9 k, -4 k, -2 k, -k, -k]] 2 PISOT(9, 81 k + 6) = [[9, 81 k + 6, 729 k + 108 k + 4, 3 2 4 3 2 6561 k + 1458 k + 108 k + 3, 59049 k + 17496 k + 1944 k + 102 k + 2, 5 4 3 2 531441 k + 196830 k + 29160 k + 2241 k + 88 k + 1], [9 k, 6 k, 4 k, 3 k, 2 k, k]] PISOT(9, 81 k + 10) = [[9, 81 k + 10], [2 + 9 k, -1 - 8 k]] PISOT(9, 81 k + 11) = [[9, 81 k + 11], [2 + 9 k, -1 - 7 k]] PISOT(9, 81 k + 12) = 2 [[9, 81 k + 12, 729 k + 216 k + 16], [9 k, 1 + 12 k, 1 + 7 k]] PISOT(9, 81 k + 13) = 2 [[9, 81 k + 13, 729 k + 234 k + 19], [1 + 9 k, 4 k, 1 + 6 k]] PISOT(9, 81 k + 16) = 2 [[9, 81 k + 16, 729 k + 288 k + 28], [2 + 9 k, -1 - 2 k, 1 + 5 k]] PISOT(9, 81 k + 17) = [ 2 3 2 [9, 81 k + 17, 729 k + 306 k + 32, 6561 k + 4131 k + 865 k + 60], [2 + 9 k, -k, -1 - 2 k, 1 + 5 k]] PISOT(9, 81 k + 20) = 2 [[9, 81 k + 20, 729 k + 360 k + 44], [2 + 9 k, 2 k, 1 + 4 k]] PISOT(9, 81 k + 21) = 2 [[9, 81 k + 21, 729 k + 378 k + 49], [3 + 9 k, -2 - 6 k, 1 + 4 k]] 2 PISOT(9, 81 k + 25) = [[9, 81 k + 25, 729 k + 450 k + 69, 3 2 6561 k + 6075 k + 1867 k + 190, 4 3 2 59049 k + 72900 k + 33642 k + 6870 k + 523], [3 + 9 k, -2 k, -2 - 6 k, k, 1 + 3 k]] PISOT(9, 81 k + 26) = 2 [[9, 81 k + 26, 729 k + 468 k + 75], [3 + 9 k, -k, -1 - 3 k]] PISOT(9, 81 k + 28) = 2 [[9, 81 k + 28, 729 k + 504 k + 87], [3 + 9 k, k, 1 + 3 k]] 2 PISOT(9, 81 k + 29) = [[9, 81 k + 29, 729 k + 522 k + 93, 3 2 6561 k + 7047 k + 2515 k + 298, 4 3 2 59049 k + 84564 k + 45306 k + 10758 k + 955], [3 + 9 k, 2 k, 2 + 6 k, k, 1 + 3 k]] 2 PISOT(9, 81 k + 31) = [[9, 81 k + 31, 729 k + 558 k + 107, 3 2 6561 k + 7533 k + 2887 k + 369, 4 3 2 59049 k + 90396 k + 51948 k + 13276 k + 1273], [3 + 9 k, 2 + 4 k, -2 - 4 k, 1 + 4 k, 2 + 5 k]] PISOT(9, 81 k + 32) = [[9, 81 k + 32], [3 + 9 k, 2 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.994621911397805 PISOT(9, 81 k + 38) = [ 2 3 2 [9, 81 k + 38, 729 k + 684 k + 160, 6561 k + 9234 k + 4324 k + 674], [4 + 9 k, 2 k, 4 + 8 k, -1 - 2 k]] PISOT(9, 81 k + 39) = [ 2 3 2 [9, 81 k + 39, 729 k + 702 k + 169, 6561 k + 9477 k + 4563 k + 732], [5 + 9 k, -3 - 6 k, k, 2 + 4 k]] PISOT(9, 81 k + 40) = [[9, 81 k + 40], [4 + 9 k, 2 + 4 k]] PISOT(9, 81 k + 41) = [[9, 81 k + 41], [5 + 9 k, -2 - 4 k]] PISOT(9, 81 k + 42) = [ 2 3 2 [9, 81 k + 42, 729 k + 756 k + 196, 6561 k + 10206 k + 5292 k + 915], [4 + 9 k, 3 + 6 k, 1 + k, -2 - 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.994616534533709 PISOT(9, 81 k + 43) = [ 2 3 2 [9, 81 k + 43, 729 k + 774 k + 205, 6561 k + 10449 k + 5539 k + 977], [5 + 9 k, -2 - 2 k, 4 + 8 k, 1 + 2 k]] PISOT(9, 81 k + 49) = [[9, 81 k + 49], [6 + 9 k, -3 - 5 k]] 2 PISOT(9, 81 k + 50) = [[9, 81 k + 50, 729 k + 900 k + 278, 3 2 6561 k + 12150 k + 7504 k + 1546, 4 3 2 59049 k + 145800 k + 135054 k + 55628 k + 8598], [6 + 9 k, -2 - 4 k, -2 - 4 k, -3 - 4 k, 3 + 5 k]] 2 PISOT(9, 81 k + 52) = [[9, 81 k + 52, 729 k + 936 k + 300, 3 2 6561 k + 12636 k + 8104 k + 1731, 4 3 2 59049 k + 151632 k + 145908 k + 62358 k + 9988], [6 + 9 k, -2 - 2 k, 4 + 6 k, -1 - k, 2 + 3 k]] PISOT(9, 81 k + 53) = 2 [[9, 81 k + 53, 729 k + 954 k + 312], [6 + 9 k, -1 - k, 2 + 3 k]] PISOT(9, 81 k + 55) = 2 [[9, 81 k + 55, 729 k + 990 k + 336], [6 + 9 k, 1 + k, -2 - 3 k]] 2 PISOT(9, 81 k + 56) = [[9, 81 k + 56, 729 k + 1008 k + 348, 3 2 6561 k + 13608 k + 9400 k + 2163, 4 3 2 59049 k + 163296 k + 169236 k + 77910 k + 13444], [6 + 9 k, 2 + 2 k, -4 - 6 k, -1 - k, 2 + 3 k]] PISOT(9, 81 k + 60) = 2 [[9, 81 k + 60, 729 k + 1080 k + 400], [6 + 9 k, 4 + 6 k, 3 + 4 k]] PISOT(9, 81 k + 61) = 2 [[9, 81 k + 61, 729 k + 1098 k + 413], [7 + 9 k, -2 - 2 k, 3 + 4 k]] PISOT(9, 81 k + 64) = [ 2 3 2 [9, 81 k + 64, 729 k + 1152 k + 455, 6561 k + 15552 k + 12286 k + 3235], [7 + 9 k, 1 + k, -1 - 2 k, -4 - 5 k]] PISOT(9, 81 k + 65) = 2 [[9, 81 k + 65, 729 k + 1170 k + 469], [7 + 9 k, 1 + 2 k, 4 + 5 k]] PISOT(9, 81 k + 68) = 2 [[9, 81 k + 68, 729 k + 1224 k + 514], [8 + 9 k, -4 - 4 k, 5 + 6 k]] PISOT(9, 81 k + 69) = 2 [[9, 81 k + 69, 729 k + 1242 k + 529], [9 + 9 k, -11 - 12 k, 6 + 7 k]] PISOT(9, 81 k + 70) = [[9, 81 k + 70], [7 + 9 k, 6 + 7 k]] PISOT(9, 81 k + 71) = [[9, 81 k + 71], [7 + 9 k, 7 + 8 k]] 2 PISOT(9, 81 k + 75) = [[9, 81 k + 75, 729 k + 1350 k + 625, 3 2 6561 k + 18225 k + 16875 k + 5208, 4 3 2 59049 k + 218700 k + 303750 k + 187494 k + 43397, 5 4 3 2 531441 k + 2460375 k + 4556250 k + 4218669 k + 1952971 k + 361617], [9 + 9 k, -6 - 6 k, 4 + 4 k, -3 - 3 k, 2 + 2 k, -1 - k]] 2 PISOT(9, 81 k + 76) = [[9, 81 k + 76, 729 k + 1368 k + 642, 3 2 6561 k + 18468 k + 17332 k + 5423, 4 3 2 59049 k + 221616 k + 311958 k + 195198 k + 45808], [8 + 9 k, 4 + 4 k, -2 - 2 k, 1 + k, -1 - k]] PISOT(9, 81 k + 77) = [ 2 3 2 [9, 81 k + 77, 729 k + 1386 k + 659, 6561 k + 18711 k + 17791 k + 5640], [9 + 9 k, -4 - 4 k, 2 + 2 k, -1 - k]] PISOT(9, 81 k + 78) = 2 [[9, 81 k + 78, 729 k + 1404 k + 676], [9 + 9 k, -3 - 3 k, 1 + k]] PISOT(9, 81 k + 79) = [[9, 81 k + 79], [9 + 9 k, -2 - 2 k]] PISOT(9, 81 k + 80) = [[9, 81 k + 80], [9 + 9 k, -1 - k]] PISOT(10, 100 k + 1) = [[10, 100 k + 1], [10 k, k]] PISOT(10, 100 k + 2) = [[10, 100 k + 2], [10 k, 2 k]] 2 PISOT(10, 100 k + 3) = [[10, 100 k + 3, 1000 k + 60 k + 1], [10 k, 3 k, k]] PISOT(10, 100 k + 4) = [ 2 3 2 [10, 100 k + 4, 1000 k + 80 k + 2, 10000 k + 1200 k + 56 k + 1], [10 k, 4 k, 2 k, k]] 2 PISOT(10, 100 k + 5) = [[10, 100 k + 5, 1000 k + 100 k + 3, 3 2 4 3 2 10000 k + 1500 k + 85 k + 2, 100000 k + 20000 k + 1650 k + 70 k + 1], [1 + 10 k, -5 k, -2 k, -k, -k]] PISOT(10, 100 k + 11) = [[10, 100 k + 11], [2 + 10 k, -1 - 9 k]] PISOT(10, 100 k + 12) = [[10, 100 k + 12], [2 + 10 k, -1 - 8 k]] PISOT(10, 100 k + 16) = [[10, 100 k + 16], [1 + 10 k, 1 + 6 k]] PISOT(10, 100 k + 24) = [[10, 100 k + 24], [2 + 10 k, 1 + 4 k]] PISOT(10, 100 k + 25) = 2 [[10, 100 k + 25, 1000 k + 500 k + 63], [3 + 10 k, -2 - 5 k, 2 + 8 k]] PISOT(10, 100 k + 26) = [[10, 100 k + 26], [3 + 10 k, -1 - 4 k]] PISOT(10, 100 k + 29) = 2 [[10, 100 k + 29, 1000 k + 580 k + 84], [3 + 10 k, -1 - k, 2 + 7 k]] 2 PISOT(10, 100 k + 32) = [[10, 100 k + 32, 1000 k + 640 k + 102, 3 2 10000 k + 9600 k + 3064 k + 325, 4 3 2 100000 k + 128000 k + 61320 k + 13028 k + 1036], [3 + 10 k, 2 k, 2 + 6 k, -k, -1 - 3 k]] PISOT(10, 100 k + 33) = [[10, 100 k + 33], [3 + 10 k, 1 + 3 k]] PISOT(10, 100 k + 34) = [[10, 100 k + 34], [4 + 10 k, -2 - 6 k]] PISOT(10, 100 k + 35) = 2 [[10, 100 k + 35, 1000 k + 700 k + 123], [4 + 10 k, -2 - 5 k, 1 + 3 k]] PISOT(10, 100 k + 38) = [[10, 100 k + 38], [3 + 10 k, 3 + 8 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.982594394352120 2 PISOT(10, 100 k + 39) = [[10, 100 k + 39, 1000 k + 780 k + 152, 3 2 10000 k + 11700 k + 4561 k + 592, 4 3 2 100000 k + 156000 k + 91230 k + 23696 k + 2306, 5 4 3 2 1000000 k + 1950000 k + 1520600 k + 592599 k + 115400 k + 8982], [5 + 10 k, -5 - 11 k, 3 + 7 k, -1 - 3 k, -1 - k, 2 + 5 k]] PISOT(10, 100 k + 41) = 2 [[10, 100 k + 41, 1000 k + 820 k + 168], [3 + 10 k, 4 + 11 k, 2 + 5 k]] PISOT(10, 100 k + 43) = [[10, 100 k + 43], [5 + 10 k, -3 - 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.972273917844570 2 PISOT(10, 100 k + 45) = [[10, 100 k + 45, 1000 k + 900 k + 203, 3 2 10000 k + 13500 k + 6085 k + 916, 4 3 2 100000 k + 180000 k + 121650 k + 36590 k + 4133, 5 4 3 2 1000000 k + 2250000 k + 2027000 k + 914025 k + 206309 k + 18648], [4 + 10 k, 2 + 5 k, 2 + 3 k, -2 - 6 k, -3 - 7 k, -1 - 2 k]] PISOT(10, 100 k + 48) = [[10, 100 k + 48], [5 + 10 k, -1 - 2 k]] PISOT(10, 100 k + 49) = 2 [[10, 100 k + 49, 1000 k + 980 k + 240], [4 + 10 k, 4 + 9 k, 2 + 4 k]] PISOT(10, 100 k + 51) = 2 [[10, 100 k + 51, 1000 k + 1020 k + 260], [6 + 10 k, -5 - 9 k, 2 + 4 k]] PISOT(10, 100 k + 52) = [[10, 100 k + 52], [5 + 10 k, 1 + 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.972273950811336 2 PISOT(10, 100 k + 55) = [[10, 100 k + 55, 1000 k + 1100 k + 303, 3 2 10000 k + 16500 k + 9085 k + 1669, 4 3 2 100000 k + 220000 k + 181650 k + 66710 k + 9193, 5 4 3 2 1000000 k + 2750000 k + 3027000 k + 1666975 k + 459259 k + 50636], [6 + 10 k, -3 - 5 k, 1 + 3 k, 4 + 6 k, -4 - 7 k, 1 + 2 k]] PISOT(10, 100 k + 57) = [[10, 100 k + 57], [5 + 10 k, 4 + 7 k]] PISOT(10, 100 k + 59) = 2 [[10, 100 k + 59, 1000 k + 1180 k + 348], [7 + 10 k, -7 - 11 k, 3 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.982594321008556 2 PISOT(10, 100 k + 61) = [[10, 100 k + 61, 1000 k + 1220 k + 372, 3 2 10000 k + 18300 k + 11161 k + 2269, 4 3 2 100000 k + 244000 k + 223230 k + 90764 k + 13840, 5 4 3 2 1000000 k + 3050000 k + 3720600 k + 2269201 k + 692002 k + 84419], [5 + 10 k, 6 + 11 k, 4 + 7 k, 2 + 3 k, -k, -3 - 5 k]] PISOT(10, 100 k + 62) = [[10, 100 k + 62], [7 + 10 k, -5 - 8 k]] PISOT(10, 100 k + 65) = 2 [[10, 100 k + 65, 1000 k + 1300 k + 423], [6 + 10 k, 3 + 5 k, 2 + 3 k]] PISOT(10, 100 k + 66) = [[10, 100 k + 66], [6 + 10 k, 4 + 6 k]] PISOT(10, 100 k + 67) = [[10, 100 k + 67], [7 + 10 k, -2 - 3 k]] 2 PISOT(10, 100 k + 68) = [[10, 100 k + 68, 1000 k + 1360 k + 462, 3 2 10000 k + 20400 k + 13864 k + 3139, 4 3 2 100000 k + 272000 k + 277320 k + 125612 k + 21328], [7 + 10 k, -2 - 2 k, 4 + 6 k, 1 + k, -2 - 3 k]] PISOT(10, 100 k + 71) = 2 [[10, 100 k + 71, 1000 k + 1420 k + 504], [7 + 10 k, k, 5 + 7 k]] PISOT(10, 100 k + 74) = [[10, 100 k + 74], [7 + 10 k, 3 + 4 k]] PISOT(10, 100 k + 75) = 2 [[10, 100 k + 75, 1000 k + 1500 k + 563], [7 + 10 k, 3 + 5 k, 6 + 8 k]] PISOT(10, 100 k + 76) = [[10, 100 k + 76], [8 + 10 k, -3 - 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.999999310740579 2 PISOT(10, 100 k + 81) = [[10, 100 k + 81, 1000 k + 1620 k + 656, 3 2 10000 k + 24300 k + 19681 k + 5313], [8 + 10 k, k, 6 + 8 k, 4 + 5 k]] PISOT(10, 100 k + 84) = [[10, 100 k + 84], [9 + 10 k, -5 - 6 k]] PISOT(10, 100 k + 88) = [[10, 100 k + 88], [8 + 10 k, 7 + 8 k]] PISOT(10, 100 k + 89) = [[10, 100 k + 89], [8 + 10 k, 8 + 9 k]] 2 PISOT(10, 100 k + 94) = [[10, 100 k + 94, 1000 k + 1880 k + 884, 3 2 10000 k + 28200 k + 26516 k + 8313, 4 3 2 100000 k + 376000 k + 530280 k + 332452 k + 78174, 5 4 3 2 1000000 k + 4700000 k + 8837600 k + 8310244 k + 3907780 k + 735135], [9 + 10 k, 4 + 4 k, -2 - 2 k, 1 + k, -1 - k, 1 + k]] 2 PISOT(10, 100 k + 95) = [[10, 100 k + 95, 1000 k + 1900 k + 903, 3 2 10000 k + 28500 k + 27085 k + 8583, 4 3 2 100000 k + 380000 k + 541650 k + 343230 k + 81581], [9 + 10 k, 5 + 5 k, -2 - 2 k, 1 + k, -1 - k]] 2 PISOT(10, 100 k + 96) = [[10, 100 k + 96, 1000 k + 1920 k + 922, 3 2 10000 k + 28800 k + 27656 k + 8855], [10 + 10 k, -4 - 4 k, 2 + 2 k, -1 - k]] PISOT(10, 100 k + 97) = 2 [[10, 100 k + 97, 1000 k + 1940 k + 941], [10 + 10 k, -3 - 3 k, 1 + k]] PISOT(10, 100 k + 98) = [[10, 100 k + 98], [10 + 10 k, -2 - 2 k]] PISOT(10, 100 k + 99) = [[10, 100 k + 99], [10 + 10 k, -1 - k]] PISOT(11, 121 k + 1) = [[11, 121 k + 1], [11 k, k]] PISOT(11, 121 k + 2) = [[11, 121 k + 2], [11 k, 2 k]] 2 PISOT(11, 121 k + 3) = [[11, 121 k + 3, 1331 k + 66 k + 1], [11 k, 3 k, k]] 2 PISOT(11, 121 k + 4) = [[11, 121 k + 4, 1331 k + 88 k + 1], [11 k, 4 k, k]] PISOT(11, 121 k + 5) = [ 2 3 2 [11, 121 k + 5, 1331 k + 110 k + 2, 14641 k + 1815 k + 69 k + 1], [1 + 11 k, -6 k, -3 k, -k]] PISOT(11, 121 k + 6) = [ 2 3 2 [11, 121 k + 6, 1331 k + 132 k + 3, 14641 k + 2178 k + 102 k + 1], [11 k, 6 k, 3 k, k]] 2 PISOT(11, 121 k + 7) = [[11, 121 k + 7, 1331 k + 154 k + 4, 3 2 4 3 2 14641 k + 2541 k + 137 k + 2, 161051 k + 37268 k + 3069 k + 100 k + 1] , [11 k, 7 k, 4 k, 2 k, k]] PISOT(11, 121 k + 12) = [[11, 121 k + 12], [2 + 11 k, -1 - 10 k]] PISOT(11, 121 k + 13) = [[11, 121 k + 13], [2 + 11 k, -1 - 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970518701889734 PISOT(11, 121 k + 15) = [ 2 3 2 [11, 121 k + 15, 1331 k + 330 k + 20, 14641 k + 5445 k + 665 k + 27], [1 + 11 k, 1 + 4 k, -6 k, -1 - 8 k]] PISOT(11, 121 k + 18) = [[11, 121 k + 18], [1 + 11 k, 1 + 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.973785918191569 2 PISOT(11, 121 k + 19) = [[11, 121 k + 19, 1331 k + 418 k + 33, 3 2 14641 k + 6897 k + 1087 k + 57, 4 3 2 161051 k + 101156 k + 23892 k + 2508 k + 98, 5 4 3 2 1771561 k + 1390895 k + 437778 k + 68932 k + 5411 k + 168], [1 + 11 k, 1 + 8 k, 3 k, 1 + 5 k, -3 k, -1 - 6 k]] 2 PISOT(11, 121 k + 20) = [[11, 121 k + 20, 1331 k + 440 k + 36, 3 2 14641 k + 7260 k + 1192 k + 65, 4 3 2 161051 k + 106480 k + 26268 k + 2870 k + 117, 5 4 3 2 1771561 k + 1464100 k + 482064 k + 79115 k + 6470 k + 211], [2 + 11 k, -1 - 2 k, 1 + 7 k, 2 k, 3 k, 1 + 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.993536596228429 PISOT(11, 121 k + 23) = [ 2 3 2 [11, 121 k + 23, 1331 k + 506 k + 48, 14641 k + 8349 k + 1585 k + 100], [1 + 11 k, 2 + 12 k, 1 + 3 k, -1 - 5 k]] PISOT(11, 121 k + 24) = [ 2 3 2 [11, 121 k + 24, 1331 k + 528 k + 52, 14641 k + 8712 k + 1720 k + 113], [3 + 11 k, -2 - 9 k, 2 k, 1 + 5 k]] PISOT(11, 121 k + 25) = [ 2 3 2 [11, 121 k + 25, 1331 k + 550 k + 57, 14641 k + 9075 k + 1879 k + 130], [2 + 11 k, 3 k, 1 + 7 k, 1 + 5 k]] PISOT(11, 121 k + 29) = [[11, 121 k + 29], [3 + 11 k, -1 - 4 k]] PISOT(11, 121 k + 30) = [[11, 121 k + 30], [2 + 11 k, 2 + 8 k]] PISOT(11, 121 k + 36) = 2 [[11, 121 k + 36, 1331 k + 792 k + 118], [3 + 11 k, 3 k, 3 + 10 k]] PISOT(11, 121 k + 39) = [ 2 3 2 [11, 121 k + 39, 1331 k + 858 k + 138, 14641 k + 14157 k + 4557 k + 488] , [4 + 11 k, -2 - 5 k, 1 + 4 k, 1 + 3 k]] PISOT(11, 121 k + 40) = 2 [[11, 121 k + 40, 1331 k + 880 k + 145], [3 + 11 k, 2 + 7 k, 1 + 3 k]] PISOT(11, 121 k + 41) = [[11, 121 k + 41], [4 + 11 k, -1 - 3 k]] 2 PISOT(11, 121 k + 42) = [[11, 121 k + 42, 1331 k + 924 k + 160, 3 2 14641 k + 15246 k + 5284 k + 610, 4 3 2 161051 k + 223608 k + 116292 k + 26860 k + 2326], [5 + 11 k, -5 - 13 k, 2 + 5 k, -1 - 2 k, 1 + 3 k]] PISOT(11, 121 k + 46) = 2 [[11, 121 k + 46, 1331 k + 1012 k + 192], [4 + 11 k, 2 k, 3 + 8 k]] PISOT(11, 121 k + 48) = 2 [[11, 121 k + 48, 1331 k + 1056 k + 209], [4 + 11 k, 2 + 4 k, -2 - 5 k]] PISOT(11, 121 k + 49) = [[11, 121 k + 49], [4 + 11 k, 2 + 5 k]] PISOT(11, 121 k + 50) = [[ 2 3 2 11, 121 k + 50, 1331 k + 1100 k + 227, 14641 k + 18150 k + 7494 k + 1031 ], [4 + 11 k, 3 + 6 k, -2 - 6 k, -2 - 5 k]] PISOT(11, 121 k + 51) = 2 [[11, 121 k + 51, 1331 k + 1122 k + 236], [6 + 11 k, -7 - 15 k, 3 + 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.991049619414240 PISOT(11, 121 k + 53) = [[ 2 3 2 11, 121 k + 53, 1331 k + 1166 k + 255, 14641 k + 19239 k + 8419 k + 1227 ], [6 + 11 k, -7 - 13 k, 7 + 14 k, -4 - 9 k]] PISOT(11, 121 k + 57) = [[ 2 3 2 11, 121 k + 57, 1331 k + 1254 k + 295, 14641 k + 20691 k + 9739 k + 1527 ], [6 + 11 k, -5 - 9 k, 4 + 8 k, -1 - 2 k]] PISOT(11, 121 k + 60) = [[11, 121 k + 60], [6 + 11 k, -3 - 6 k]] PISOT(11, 121 k + 61) = [[11, 121 k + 61], [5 + 11 k, 3 + 6 k]] 2 PISOT(11, 121 k + 64) = [[11, 121 k + 64, 1331 k + 1408 k + 372, 3 2 14641 k + 23232 k + 12280 k + 2162], [5 + 11 k, 4 + 9 k, 4 + 8 k, 1 + 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.991046332718422 2 PISOT(11, 121 k + 68) = [[11, 121 k + 68, 1331 k + 1496 k + 420, 3 2 14641 k + 24684 k + 13864 k + 2594], [5 + 11 k, 6 + 13 k, 7 + 14 k, 5 + 9 k]] PISOT(11, 121 k + 70) = 2 [[11, 121 k + 70, 1331 k + 1540 k + 445], [5 + 11 k, 8 + 15 k, 4 + 7 k]] 2 PISOT(11, 121 k + 71) = [[11, 121 k + 71, 1331 k + 1562 k + 458, 3 2 14641 k + 25773 k + 15117 k + 2954], [7 + 11 k, -3 - 6 k, -4 - 6 k, 3 + 5 k]] PISOT(11, 121 k + 72) = [[11, 121 k + 72], [7 + 11 k, -3 - 5 k]] PISOT(11, 121 k + 73) = 2 [[11, 121 k + 73, 1331 k + 1606 k + 484], [7 + 11 k, -2 - 4 k, -3 - 5 k]] PISOT(11, 121 k + 75) = 2 [[11, 121 k + 75, 1331 k + 1650 k + 511], [7 + 11 k, -2 - 2 k, 5 + 8 k]] 2 PISOT(11, 121 k + 79) = [[11, 121 k + 79, 1331 k + 1738 k + 567, 3 2 14641 k + 28677 k + 18715 k + 4069, 4 3 2 161051 k + 420596 k + 411774 k + 179104 k + 29201], [6 + 11 k, 8 + 13 k, 3 + 5 k, 1 + 2 k, 2 + 3 k]] PISOT(11, 121 k + 80) = [[11, 121 k + 80], [7 + 11 k, 2 + 3 k]] PISOT(11, 121 k + 81) = 2 [[11, 121 k + 81, 1331 k + 1782 k + 596], [8 + 11 k, -5 - 7 k, 2 + 3 k]] 2 PISOT(11, 121 k + 82) = [[11, 121 k + 82, 1331 k + 1804 k + 611, 3 2 14641 k + 29766 k + 20166 k + 4553], [7 + 11 k, 3 + 5 k, 3 + 4 k, -2 - 3 k]] PISOT(11, 121 k + 85) = 2 [[11, 121 k + 85, 1331 k + 1870 k + 657], [8 + 11 k, -3 - 3 k, 7 + 10 k]] PISOT(11, 121 k + 91) = [[11, 121 k + 91], [9 + 11 k, -6 - 8 k]] PISOT(11, 121 k + 92) = [[11, 121 k + 92], [8 + 11 k, 3 + 4 k]] 2 PISOT(11, 121 k + 96) = [[11, 121 k + 96, 1331 k + 2112 k + 838, 3 2 14641 k + 34848 k + 27652 k + 7315], [9 + 11 k, -3 - 3 k, 6 + 7 k, -4 - 5 k]] 2 PISOT(11, 121 k + 97) = [[11, 121 k + 97, 1331 k + 2134 k + 855, 3 2 14641 k + 35211 k + 28219 k + 7536], [8 + 11 k, 7 + 9 k, 2 + 2 k, -4 - 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.993533177467672 2 PISOT(11, 121 k + 98) = [[11, 121 k + 98, 1331 k + 2156 k + 873, 3 2 14641 k + 35574 k + 28810 k + 7777], [10 + 11 k, -10 - 12 k, 2 + 3 k, 4 + 5 k]] 2 PISOT(11, 121 k + 101) = [[11, 121 k + 101, 1331 k + 2222 k + 927, 3 2 14641 k + 36663 k + 30595 k + 8508, 4 3 2 161051 k + 537724 k + 673134 k + 374430 k + 78086, 5 4 3 2 1771561 k + 7393705 k + 12341274 k + 10298087 k + 4295837 k + 716669], [9 + 11 k, 1 + 2 k, 6 + 7 k, -2 - 2 k, 3 + 3 k, -5 - 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.973784811446376 2 PISOT(11, 121 k + 102) = [[11, 121 k + 102, 1331 k + 2244 k + 946, 3 2 14641 k + 37026 k + 31216 k + 8774, 4 3 2 161051 k + 543048 k + 686730 k + 386012 k + 81377, 5 4 3 2 1771561 k + 7466910 k + 12589808 k + 10614642 k + 4475106 k + 754755], [10 + 11 k, -7 - 8 k, 3 + 3 k, -4 - 5 k, -3 - 3 k, 5 + 6 k]] PISOT(11, 121 k + 103) = [[11, 121 k + 103], [10 + 11 k, -6 - 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970516806497525 2 PISOT(11, 121 k + 106) = [[11, 121 k + 106, 1331 k + 2332 k + 1021, 3 2 14641 k + 38478 k + 33698 k + 9834], [10 + 11 k, -3 - 4 k, -6 - 6 k, 7 + 8 k]] PISOT(11, 121 k + 108) = [[11, 121 k + 108], [9 + 11 k, 8 + 9 k]] PISOT(11, 121 k + 109) = [[11, 121 k + 109], [9 + 11 k, 9 + 10 k]] 2 PISOT(11, 121 k + 114) = [[11, 121 k + 114, 1331 k + 2508 k + 1181, 3 2 14641 k + 41382 k + 38978 k + 12235, 4 3 2 161051 k + 606936 k + 857571 k + 538438 k + 126753], [11 + 11 k, -7 - 7 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(11, 121 k + 115) = [[11, 121 k + 115, 1331 k + 2530 k + 1202, 3 2 14641 k + 41745 k + 39669 k + 12564], [11 + 11 k, -6 - 6 k, 3 + 3 k, -1 - k]] 2 PISOT(11, 121 k + 116) = [[11, 121 k + 116, 1331 k + 2552 k + 1223, 3 2 14641 k + 42108 k + 40362 k + 12894], [10 + 11 k, 6 + 6 k, -3 - 3 k, 1 + k]] PISOT(11, 121 k + 117) = 2 [[11, 121 k + 117, 1331 k + 2574 k + 1244], [11 + 11 k, -4 - 4 k, 1 + k]] PISOT(11, 121 k + 118) = 2 [[11, 121 k + 118, 1331 k + 2596 k + 1266], [11 + 11 k, -3 - 3 k, 1 + k]] PISOT(11, 121 k + 119) = [[11, 121 k + 119], [11 + 11 k, -2 - 2 k]] PISOT(11, 121 k + 120) = [[11, 121 k + 120], [11 + 11 k, -1 - k]] PISOT(12, 144 k + 1) = [[12, 144 k + 1], [12 k, k]] PISOT(12, 144 k + 2) = [[12, 144 k + 2], [12 k, 2 k]] 2 PISOT(12, 144 k + 3) = [[12, 144 k + 3, 1728 k + 72 k + 1], [12 k, 3 k, k]] 2 PISOT(12, 144 k + 4) = [[12, 144 k + 4, 1728 k + 96 k + 1], [12 k, 4 k, k]] PISOT(12, 144 k + 5) = [ 2 3 2 [12, 144 k + 5, 1728 k + 120 k + 2, 20736 k + 2160 k + 73 k + 1], [12 k, 5 k, 2 k, k]] 2 PISOT(12, 144 k + 6) = [[12, 144 k + 6, 1728 k + 144 k + 3, 3 2 4 3 2 20736 k + 2592 k + 108 k + 2, 248832 k + 41472 k + 2592 k + 84 k + 1], [1 + 12 k, -6 k, -3 k, -k, -k]] 2 PISOT(12, 144 k + 7) = [[12, 144 k + 7, 1728 k + 168 k + 4, 3 2 4 3 2 20736 k + 3024 k + 145 k + 2, 248832 k + 48384 k + 3492 k + 104 k + 1] , [12 k, 7 k, 4 k, 2 k, k]] 2 PISOT(12, 144 k + 8) = [[12, 144 k + 8, 1728 k + 192 k + 5, 3 2 4 3 2 20736 k + 3456 k + 184 k + 3, 248832 k + 55296 k + 4464 k + 152 k + 2, 5 4 3 2 2985984 k + 829440 k + 89856 k + 4688 k + 121 k + 1], [12 k, 8 k, 5 k, 3 k, 2 k, k]] PISOT(12, 144 k + 13) = [[12, 144 k + 13], [2 + 12 k, -1 - 11 k]] PISOT(12, 144 k + 14) = [[12, 144 k + 14], [2 + 12 k, -1 - 10 k]] PISOT(12, 144 k + 16) = 2 [[12, 144 k + 16, 1728 k + 384 k + 21], [12 k, 1 + 16 k, 1 + 9 k]] PISOT(12, 144 k + 20) = 2 [[12, 144 k + 20, 1728 k + 480 k + 33], [2 + 12 k, -4 k, -1 - 7 k]] PISOT(12, 144 k + 21) = 2 [[12, 144 k + 21, 1728 k + 504 k + 37], [2 + 12 k, -1 - 3 k, 1 + 7 k]] PISOT(12, 144 k + 23) = [ 2 3 2 [12, 144 k + 23, 1728 k + 552 k + 44, 20736 k + 9936 k + 1585 k + 84], [1 + 12 k, 2 + 11 k, -3 k, -1 - 6 k]] PISOT(12, 144 k + 28) = 2 [[12, 144 k + 28, 1728 k + 672 k + 65], [3 + 12 k, -2 - 8 k, 1 + 5 k]] PISOT(12, 144 k + 29) = [[12, 144 k + 29], [2 + 12 k, 1 + 5 k]] PISOT(12, 144 k + 30) = [ 2 3 2 [12, 144 k + 30, 1728 k + 720 k + 75, 20736 k + 12960 k + 2700 k + 188], [3 + 12 k, -1 - 6 k, -1 - 3 k, 1 + 5 k]] PISOT(12, 144 k + 32) = 2 [[12, 144 k + 32, 1728 k + 768 k + 85], [2 + 12 k, 1 + 8 k, 2 + 9 k]] 2 PISOT(12, 144 k + 34) = [[12, 144 k + 34, 1728 k + 816 k + 96, 3 2 20736 k + 14688 k + 3460 k + 271, 4 3 2 248832 k + 235008 k + 83088 k + 13032 k + 765], [3 + 12 k, -1 - 2 k, 2 + 6 k, -2 - 7 k, 1 + 4 k]] PISOT(12, 144 k + 37) = 2 [[12, 144 k + 37, 1728 k + 888 k + 114], [2 + 12 k, 3 + 13 k, 1 + 4 k]] 2 PISOT(12, 144 k + 38) = [[12, 144 k + 38, 1728 k + 912 k + 120, 3 2 20736 k + 16416 k + 4324 k + 379, 4 3 2 248832 k + 262656 k + 103824 k + 18216 k + 1197], [3 + 12 k, 1 + 2 k, -2 - 6 k, 1 + 5 k, 1 + 4 k]] PISOT(12, 144 k + 41) = [[12, 144 k + 41], [4 + 12 k, -2 - 7 k]] 2 PISOT(12, 144 k + 42) = [[12, 144 k + 42, 1728 k + 1008 k + 147, 3 2 20736 k + 18144 k + 5292 k + 515, 4 3 2 248832 k + 290304 k + 127008 k + 24708 k + 1804], [3 + 12 k, 2 + 6 k, -1 - 3 k, 2 k, 2 + 7 k]] PISOT(12, 144 k + 49) = 2 [[12, 144 k + 49, 1728 k + 1176 k + 200], [5 + 12 k, -4 - 11 k, 1 + 3 k]] PISOT(12, 144 k + 58) = [[12, 144 k + 58], [4 + 12 k, 4 + 10 k]] 2 PISOT(12, 144 k + 59) = [[12, 144 k + 59, 1728 k + 1416 k + 290, 3 2 20736 k + 25488 k + 10441 k + 1425], [4 + 12 k, 4 + 11 k, 2 + 6 k, 2 + 5 k]] PISOT(12, 144 k + 61) = 2 [[12, 144 k + 61, 1728 k + 1464 k + 310], [5 + 12 k, 1 + k, -3 - 7 k]] PISOT(12, 144 k + 64) = 2 [[12, 144 k + 64, 1728 k + 1536 k + 341], [5 + 12 k, 1 + 4 k, 4 + 9 k]] 2 PISOT(12, 144 k + 68) = [[12, 144 k + 68, 1728 k + 1632 k + 385, 3 2 20736 k + 29376 k + 13864 k + 2180, 4 3 2 248832 k + 470016 k + 332784 k + 104680 k + 12344], [6 + 12 k, -2 - 4 k, k, 3 + 6 k, -1 - 2 k]] PISOT(12, 144 k + 70) = [[12, 144 k + 70], [6 + 12 k, -1 - 2 k]] PISOT(12, 144 k + 71) = 2 [[12, 144 k + 71, 1728 k + 1704 k + 420], [6 + 12 k, -1 - k, 3 + 6 k]] PISOT(12, 144 k + 73) = 2 [[12, 144 k + 73, 1728 k + 1752 k + 444], [6 + 12 k, k, 3 + 6 k]] PISOT(12, 144 k + 74) = [[12, 144 k + 74], [6 + 12 k, 1 + 2 k]] 2 PISOT(12, 144 k + 76) = [[12, 144 k + 76, 1728 k + 1824 k + 481, 3 2 20736 k + 32832 k + 17320 k + 3044, 4 3 2 248832 k + 525312 k + 415728 k + 146168 k + 19264], [6 + 12 k, 2 + 4 k, 1 + k, -3 - 6 k, -1 - 2 k]] PISOT(12, 144 k + 80) = 2 [[12, 144 k + 80, 1728 k + 1920 k + 533], [7 + 12 k, -3 - 4 k, 5 + 9 k]] PISOT(12, 144 k + 83) = 2 [[12, 144 k + 83, 1728 k + 1992 k + 574], [7 + 12 k, -k, -4 - 7 k]] 2 PISOT(12, 144 k + 85) = [[12, 144 k + 85, 1728 k + 2040 k + 602, 3 2 20736 k + 36720 k + 21673 k + 4264], [8 + 12 k, -7 - 11 k, 4 + 6 k, -3 - 5 k]] PISOT(12, 144 k + 86) = [[12, 144 k + 86], [8 + 12 k, -6 - 10 k]] 2 PISOT(12, 144 k + 90) = [[12, 144 k + 90, 1728 k + 2160 k + 675, 3 2 20736 k + 38880 k + 24300 k + 5063], [7 + 12 k, 3 + 6 k, 5 + 9 k, 5 + 8 k]] PISOT(12, 144 k + 95) = 2 [[12, 144 k + 95, 1728 k + 2280 k + 752], [7 + 12 k, 7 + 11 k, 2 + 3 k]] 2 PISOT(12, 144 k + 102) = [[12, 144 k + 102, 1728 k + 2448 k + 867, 3 2 20736 k + 44064 k + 31212 k + 7370, 4 3 2 248832 k + 705024 k + 749088 k + 353748 k + 62649, 5 4 3 2 2985984 k + 10575360 k + 14981760 k + 10612296 k + 3758745 k + 532551], [8 + 12 k, 4 + 6 k, 2 + 3 k, 1 + 2 k, 3 + 5 k, 5 + 7 k]] PISOT(12, 144 k + 103) = [[12, 144 k + 103], [8 + 12 k, 5 + 7 k]] 2 PISOT(12, 144 k + 106) = [[12, 144 k + 106, 1728 k + 2544 k + 936, 3 2 20736 k + 45792 k + 33700 k + 8265, 4 3 2 248832 k + 732672 k + 808848 k + 396792 k + 72981], [9 + 12 k, -1 - 2 k, -4 - 6 k, -4 - 5 k, 3 + 4 k]] PISOT(12, 144 k + 107) = [ 2 [12, 144 k + 107, 1728 k + 2568 k + 954], [10 + 12 k, -10 - 13 k, 3 + 4 k] ] 2 PISOT(12, 144 k + 110) = [[12, 144 k + 110, 1728 k + 2640 k + 1008, 3 2 20736 k + 47520 k + 36292 k + 9237, 4 3 2 248832 k + 760320 k + 871056 k + 443448 k + 84645], [9 + 12 k, 1 + 2 k, 4 + 6 k, 5 + 7 k, 3 + 4 k]] PISOT(12, 144 k + 112) = [ 2 [12, 144 k + 112, 1728 k + 2688 k + 1045], [10 + 12 k, -7 - 8 k, 7 + 9 k]] 2 PISOT(12, 144 k + 114) = [[12, 144 k + 114, 1728 k + 2736 k + 1083, 3 2 20736 k + 49248 k + 38988 k + 10289, 4 3 2 248832 k + 787968 k + 935712 k + 493860 k + 97750], [10 + 12 k, -5 - 6 k, 3 + 3 k, -6 - 7 k, 4 + 5 k]] PISOT(12, 144 k + 115) = [[12, 144 k + 115], [10 + 12 k, -4 - 5 k]] PISOT(12, 144 k + 116) = 2 [[12, 144 k + 116, 1728 k + 2784 k + 1121], [9 + 12 k, 6 + 8 k, 4 + 5 k]] 2 PISOT(12, 144 k + 121) = [[12, 144 k + 121, 1728 k + 2904 k + 1220, 3 2 20736 k + 52272 k + 43921 k + 12301], [11 + 12 k, -9 - 11 k, -3 - 3 k, 5 + 6 k]] PISOT(12, 144 k + 123) = 2 [[12, 144 k + 123, 1728 k + 2952 k + 1261], [10 + 12 k, 2 + 3 k, 6 + 7 k]] PISOT(12, 144 k + 124) = [ 2 [12, 144 k + 124, 1728 k + 2976 k + 1281], [10 + 12 k, 4 + 4 k, -6 - 7 k]] 2 PISOT(12, 144 k + 126) = [[12, 144 k + 126, 1728 k + 3024 k + 1323, 3 2 20736 k + 54432 k + 47628 k + 13892], [10 + 12 k, 5 + 6 k, 2 + 3 k, 7 + 8 k]] 2 PISOT(12, 144 k + 128) = [[12, 144 k + 128, 1728 k + 3072 k + 1365], [12 + 12 k, -15 - 16 k, 8 + 9 k]] PISOT(12, 144 k + 130) = [[12, 144 k + 130], [10 + 12 k, 9 + 10 k]] PISOT(12, 144 k + 131) = [[12, 144 k + 131], [10 + 12 k, 10 + 11 k]] 2 PISOT(12, 144 k + 136) = [[12, 144 k + 136, 1728 k + 3264 k + 1541, 3 2 20736 k + 58752 k + 55480 k + 17461, 4 3 2 248832 k + 940032 k + 1331568 k + 838216 k + 197850, 5 4 3 2 2985984 k + 14100480 k + 26631936 k + 25148080 k + 11872473 k + 2241832 ], [12 + 12 k, -8 - 8 k, 5 + 5 k, -3 - 3 k, 2 + 2 k, -1 - k]] 2 PISOT(12, 144 k + 137) = [[12, 144 k + 137, 1728 k + 3288 k + 1564, 3 2 20736 k + 59184 k + 56305 k + 17855, 4 3 2 248832 k + 946944 k + 1351332 k + 857056 k + 203837], [12 + 12 k, -7 - 7 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(12, 144 k + 138) = [[12, 144 k + 138, 1728 k + 3312 k + 1587, 3 2 20736 k + 59616 k + 57132 k + 18251], [12 + 12 k, -6 - 6 k, 3 + 3 k, -1 - k]] 2 PISOT(12, 144 k + 139) = [[12, 144 k + 139, 1728 k + 3336 k + 1610, 3 2 20736 k + 60048 k + 57961 k + 18648], [12 + 12 k, -5 - 5 k, 2 + 2 k, -1 - k]] PISOT(12, 144 k + 140) = 2 [[12, 144 k + 140, 1728 k + 3360 k + 1633], [12 + 12 k, -4 - 4 k, 1 + k]] PISOT(12, 144 k + 141) = 2 [[12, 144 k + 141, 1728 k + 3384 k + 1657], [12 + 12 k, -3 - 3 k, 1 + k]] PISOT(12, 144 k + 142) = [[12, 144 k + 142], [12 + 12 k, -2 - 2 k]] PISOT(12, 144 k + 143) = [[12, 144 k + 143], [12 + 12 k, -1 - k]] PISOT(13, 169 k + 1) = [[13, 169 k + 1], [13 k, k]] PISOT(13, 169 k + 2) = [[13, 169 k + 2], [13 k, 2 k]] 2 PISOT(13, 169 k + 3) = [[13, 169 k + 3, 2197 k + 78 k + 1], [13 k, 3 k, k]] 2 PISOT(13, 169 k + 4) = [[13, 169 k + 4, 2197 k + 104 k + 1], [13 k, 4 k, k]] PISOT(13, 169 k + 5) = [ 2 3 2 [13, 169 k + 5, 2197 k + 130 k + 2, 28561 k + 2535 k + 77 k + 1], [13 k, 5 k, 2 k, k]] PISOT(13, 169 k + 6) = [ 2 3 2 [13, 169 k + 6, 2197 k + 156 k + 3, 28561 k + 3042 k + 114 k + 1], [13 k, 6 k, 3 k, k]] 2 PISOT(13, 169 k + 7) = [[13, 169 k + 7, 2197 k + 182 k + 4, 3 2 4 3 2 28561 k + 3549 k + 153 k + 2, 371293 k + 61516 k + 3939 k + 108 k + 1] , [13 k, 7 k, 4 k, 2 k, k]] 2 PISOT(13, 169 k + 8) = [[13, 169 k + 8, 2197 k + 208 k + 5, 3 2 4 3 2 28561 k + 4056 k + 194 k + 3, 371293 k + 70304 k + 5031 k + 158 k + 2, 5 4 3 2 4826809 k + 1142440 k + 108836 k + 5153 k + 125 k + 1], [13 k, 8 k, 5 k, 3 k, 2 k, k]] PISOT(13, 169 k + 14) = [[13, 169 k + 14], [2 + 13 k, -1 - 12 k]] PISOT(13, 169 k + 15) = [[13, 169 k + 15], [2 + 13 k, -1 - 11 k]] PISOT(13, 169 k + 19) = 2 [[13, 169 k + 19, 2197 k + 494 k + 28], [1 + 13 k, 6 k, 1 + 9 k]] PISOT(13, 169 k + 21) = [[13, 169 k + 21], [1 + 13 k, 1 + 8 k]] 2 PISOT(13, 169 k + 22) = [[13, 169 k + 22, 2197 k + 572 k + 37, 3 2 28561 k + 11154 k + 1446 k + 62, 4 3 2 371293 k + 193336 k + 37635 k + 3240 k + 104], [2 + 13 k, -1 - 4 k, 1 + 6 k, -1 - 3 k, 1 + 8 k]] 2 PISOT(13, 169 k + 23) = [[13, 169 k + 23, 2197 k + 598 k + 41, 3 2 28561 k + 11661 k + 1595 k + 73, 4 3 2 371293 k + 202124 k + 41418 k + 3784 k + 130], [2 + 13 k, -3 k, -1 - 5 k, 4 k, 1 + 7 k]] PISOT(13, 169 k + 24) = 2 [[13, 169 k + 24, 2197 k + 624 k + 44], [1 + 13 k, 1 + 11 k, 1 + 7 k]] PISOT(13, 169 k + 28) = 2 [[13, 169 k + 28, 2197 k + 728 k + 60], [1 + 13 k, 2 + 15 k, 1 + 6 k]] PISOT(13, 169 k + 29) = 2 [[13, 169 k + 29, 2197 k + 754 k + 65], [2 + 13 k, 1 + 3 k, -1 - 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.985649982540292 2 PISOT(13, 169 k + 32) = [[13, 169 k + 32, 2197 k + 832 k + 79, 3 2 28561 k + 16224 k + 3078 k + 195, 4 3 2 371293 k + 281216 k + 79989 k + 10126 k + 481, 5 4 3 2 4826809 k + 4569760 k + 1732588 k + 328817 k + 31227 k + 1186], [2 + 13 k, 1 + 6 k, 1 + 2 k, -1 - 8 k, -1 - 7 k, -1 - 5 k]] PISOT(13, 169 k + 33) = 2 [[13, 169 k + 33, 2197 k + 858 k + 84], [2 + 13 k, 1 + 7 k, 1 + 5 k]] PISOT(13, 169 k + 34) = [[13, 169 k + 34], [3 + 13 k, -1 - 5 k]] 2 PISOT(13, 169 k + 35) = [[13, 169 k + 35, 2197 k + 910 k + 94, 3 2 28561 k + 17745 k + 3669 k + 252, 4 3 2 371293 k + 307580 k + 95433 k + 13132 k + 676], [3 + 13 k, -1 - 4 k, 1 + 2 k, -2 - 8 k, 1 + 5 k]] PISOT(13, 169 k + 37) = 2 [[13, 169 k + 37, 2197 k + 962 k + 105], [4 + 13 k, -4 - 15 k, 2 + 9 k]] PISOT(13, 169 k + 38) = 2 [[13, 169 k + 38, 2197 k + 988 k + 111], [2 + 13 k, 2 + 12 k, 2 + 9 k]] PISOT(13, 169 k + 41) = 2 [[13, 169 k + 41, 2197 k + 1066 k + 129], [4 + 13 k, -3 - 11 k, 1 + 4 k]] PISOT(13, 169 k + 43) = [[13, 169 k + 43], [3 + 13 k, 1 + 4 k]] PISOT(13, 169 k + 44) = [[ 2 3 2 13, 169 k + 44, 2197 k + 1144 k + 149, 28561 k + 22308 k + 5810 k + 505] , [3 + 13 k, 2 + 5 k, -2 - 9 k, -1 - 4 k]] PISOT(13, 169 k + 46) = 2 [[13, 169 k + 46, 2197 k + 1196 k + 163], [5 + 13 k, -6 - 19 k, 3 + 11 k]] PISOT(13, 169 k + 48) = 2 [[13, 169 k + 48, 2197 k + 1248 k + 177], [3 + 13 k, 2 + 9 k, 2 + 7 k]] PISOT(13, 169 k + 51) = 2 [[13, 169 k + 51, 2197 k + 1326 k + 200], [5 + 13 k, -5 - 14 k, 3 + 10 k]] PISOT(13, 169 k + 55) = [[13, 169 k + 55], [4 + 13 k, 1 + 3 k]] PISOT(13, 169 k + 56) = [[13, 169 k + 56], [5 + 13 k, -3 - 9 k]] PISOT(13, 169 k + 57) = 2 [[13, 169 k + 57, 2197 k + 1482 k + 250], [4 + 13 k, 1 + 5 k, 3 + 9 k]] 2 PISOT(13, 169 k + 58) = [[13, 169 k + 58, 2197 k + 1508 k + 259, 3 2 28561 k + 29406 k + 10098 k + 1157], [5 + 13 k, -3 - 7 k, 3 + 8 k, -1 - 3 k]] 2 PISOT(13, 169 k + 59) = [[13, 169 k + 59, 2197 k + 1534 k + 268, 3 2 28561 k + 29913 k + 10449 k + 1217, 4 3 2 371293 k + 518492 k + 271635 k + 63266 k + 5526], [5 + 13 k, -2 - 6 k, -1 - k, 3 + 8 k, -1 - 3 k]] PISOT(13, 169 k + 61) = 2 [[13, 169 k + 61, 2197 k + 1586 k + 286], [6 + 13 k, -7 - 17 k, 4 + 11 k]] PISOT(13, 169 k + 64) = 2 [[13, 169 k + 64, 2197 k + 1664 k + 315], [5 + 13 k, -1 - k, 3 + 8 k]] 2 PISOT(13, 169 k + 66) = [[13, 169 k + 66, 2197 k + 1716 k + 335, 3 2 28561 k + 33462 k + 13066 k + 1700, 4 3 2 371293 k + 580008 k + 339729 k + 88420 k + 8627], [5 + 13 k, k, 2 + 5 k, -k, -2 - 5 k]] PISOT(13, 169 k + 68) = [[13, 169 k + 68], [6 + 13 k, -4 - 10 k]] 2 PISOT(13, 169 k + 69) = [[13, 169 k + 69, 2197 k + 1794 k + 366, 3 2 28561 k + 34983 k + 14277 k + 1941], [6 + 13 k, -4 - 9 k, 2 + 4 k, -2 - 5 k]] 2 PISOT(13, 169 k + 71) = [[13, 169 k + 71, 2197 k + 1846 k + 388, 3 2 28561 k + 35997 k + 15129 k + 2120, 4 3 2 371293 k + 623948 k + 393315 k + 110216 k + 11584], [5 + 13 k, 3 + 6 k, -3 - 6 k, 2 + 6 k, 3 + 7 k]] PISOT(13, 169 k + 72) = [[13, 169 k + 72], [5 + 13 k, 3 + 7 k]] PISOT(13, 169 k + 75) = 2 [[13, 169 k + 75, 2197 k + 1950 k + 433], [6 + 13 k, -2 - 3 k, 4 + 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.997296608670196 2 PISOT(13, 169 k + 82) = [[13, 169 k + 82, 2197 k + 2132 k + 517, 3 2 28561 k + 41574 k + 20166 k + 3260], [6 + 13 k, 1 + 4 k, 6 + 12 k, -1 - 2 k]] 2 PISOT(13, 169 k + 83) = [[13, 169 k + 83, 2197 k + 2158 k + 530, 3 2 28561 k + 42081 k + 20669 k + 3384], [7 + 13 k, -4 - 8 k, k, 3 + 6 k]] PISOT(13, 169 k + 84) = [[13, 169 k + 84], [6 + 13 k, 3 + 6 k]] PISOT(13, 169 k + 85) = [[13, 169 k + 85], [7 + 13 k, -3 - 6 k]] 2 PISOT(13, 169 k + 86) = [[13, 169 k + 86, 2197 k + 2236 k + 569, 3 2 28561 k + 43602 k + 22190 k + 3765], [6 + 13 k, 4 + 8 k, 1 + k, -3 - 6 k] ] Take it with a grain of salt, for k=1000 the Pisot index is, 0.997293845746745 2 PISOT(13, 169 k + 87) = [[13, 169 k + 87, 2197 k + 2262 k + 582, 3 2 28561 k + 44109 k + 22701 k + 3893], [7 + 13 k, -3 - 4 k, 6 + 12 k, 1 + 2 k]] PISOT(13, 169 k + 94) = 2 [[13, 169 k + 94, 2197 k + 2444 k + 680], [7 + 13 k, 1 + 3 k, 5 + 9 k]] PISOT(13, 169 k + 97) = [[13, 169 k + 97], [8 + 13 k, -4 - 7 k]] 2 PISOT(13, 169 k + 98) = [[13, 169 k + 98, 2197 k + 2548 k + 739, 3 2 28561 k + 49686 k + 28818 k + 5573, 4 3 2 371293 k + 861224 k + 749229 k + 289742 k + 42028], [8 + 13 k, -3 - 6 k, -3 - 6 k, -4 - 6 k, 4 + 7 k]] 2 PISOT(13, 169 k + 100) = [[13, 169 k + 100, 2197 k + 2600 k + 769, 3 2 28561 k + 50700 k + 29994 k + 5914], [7 + 13 k, 5 + 9 k, 2 + 4 k, 3 + 5 k]] PISOT(13, 169 k + 101) = [[13, 169 k + 101], [7 + 13 k, 6 + 10 k]] 2 PISOT(13, 169 k + 103) = [[13, 169 k + 103, 2197 k + 2678 k + 816, 3 2 28561 k + 52221 k + 31825 k + 6465, 4 3 2 371293 k + 905164 k + 827463 k + 336186 k + 51221], [8 + 13 k, -1 - k, 3 + 5 k, 1 + k, -3 - 5 k]] PISOT(13, 169 k + 105) = 2 [[13, 169 k + 105, 2197 k + 2730 k + 848], [8 + 13 k, k, 5 + 8 k]] PISOT(13, 169 k + 108) = [ 2 [13, 169 k + 108, 2197 k + 2808 k + 897], [7 + 13 k, 10 + 17 k, 7 + 11 k]] 2 PISOT(13, 169 k + 110) = [[13, 169 k + 110, 2197 k + 2860 k + 931, 3 2 28561 k + 55770 k + 36306 k + 7880, 4 3 2 371293 k + 966680 k + 943917 k + 409700 k + 66696], [8 + 13 k, 4 + 6 k, -k, -5 - 8 k, -2 - 3 k]] 2 PISOT(13, 169 k + 111) = [[13, 169 k + 111, 2197 k + 2886 k + 948, 3 2 28561 k + 56277 k + 36969 k + 8096], [8 + 13 k, 4 + 7 k, 5 + 8 k, 2 + 3 k]] PISOT(13, 169 k + 112) = 2 [[13, 169 k + 112, 2197 k + 2912 k + 965], [9 + 13 k, -4 - 5 k, 6 + 9 k]] PISOT(13, 169 k + 113) = [[13, 169 k + 113], [8 + 13 k, 6 + 9 k]] PISOT(13, 169 k + 114) = [[13, 169 k + 114], [9 + 13 k, -2 - 3 k]] PISOT(13, 169 k + 118) = [ 2 [13, 169 k + 118, 2197 k + 3068 k + 1071], [8 + 13 k, 9 + 14 k, 7 + 10 k]] PISOT(13, 169 k + 121) = [ 2 [13, 169 k + 121, 2197 k + 3146 k + 1126], [10 + 13 k, -7 - 9 k, 5 + 7 k]] PISOT(13, 169 k + 123) = [ 2 [13, 169 k + 123, 2197 k + 3198 k + 1164], [8 + 13 k, 13 + 19 k, 8 + 11 k] ] 2 PISOT(13, 169 k + 125) = [[13, 169 k + 125, 2197 k + 3250 k + 1202, 3 2 28561 k + 63375 k + 46877 k + 11558], [10 + 13 k, -3 - 5 k, -7 - 9 k, 3 + 4 k]] PISOT(13, 169 k + 126) = [[13, 169 k + 126], [10 + 13 k, -3 - 4 k]] PISOT(13, 169 k + 128) = 2 [[13, 169 k + 128, 2197 k + 3328 k + 1260], [9 + 13 k, 8 + 11 k, 3 + 4 k]] 2 PISOT(13, 169 k + 131) = [[13, 169 k + 131, 2197 k + 3406 k + 1320], [11 + 13 k, -10 - 12 k, 7 + 9 k]] PISOT(13, 169 k + 132) = [ 2 [13, 169 k + 132, 2197 k + 3432 k + 1340], [9 + 13 k, 11 + 15 k, 7 + 9 k]] 2 PISOT(13, 169 k + 134) = [[13, 169 k + 134, 2197 k + 3484 k + 1381, 3 2 28561 k + 67938 k + 53862 k + 14233, 4 3 2 371293 k + 1177592 k + 1400451 k + 740166 k + 146690], [10 + 13 k, 3 + 4 k, 1 + 2 k, 6 + 8 k, 4 + 5 k]] PISOT(13, 169 k + 135) = [[13, 169 k + 135], [10 + 13 k, 4 + 5 k]] PISOT(13, 169 k + 136) = [ 2 [13, 169 k + 136, 2197 k + 3536 k + 1423], [11 + 13 k, -6 - 7 k, 4 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.985646655704477 2 PISOT(13, 169 k + 137) = [[13, 169 k + 137, 2197 k + 3562 k + 1444, 3 2 28561 k + 69459 k + 56313 k + 15220, 4 3 2 371293 k + 1203956 k + 1464099 k + 791376 k + 160421, 5 4 3 2 4826809 k + 19564285 k + 31721638 k + 25718477 k + 10426362 k + 1690861 ], [11 + 13 k, -5 - 6 k, 1 + 2 k, 7 + 8 k, -6 - 7 k, 4 + 5 k]] PISOT(13, 169 k + 140) = [ 2 [13, 169 k + 140, 2197 k + 3640 k + 1508], [11 + 13 k, -2 - 3 k, -5 - 6 k] ] 2 PISOT(13, 169 k + 141) = [[13, 169 k + 141, 2197 k + 3666 k + 1529], [12 + 13 k, -13 - 15 k, 5 + 6 k]] 2 PISOT(13, 169 k + 145) = [[13, 169 k + 145, 2197 k + 3770 k + 1617], [12 + 13 k, -10 - 11 k, 6 + 7 k]] 2 PISOT(13, 169 k + 146) = [[13, 169 k + 146, 2197 k + 3796 k + 1640, 3 2 28561 k + 74022 k + 63956 k + 18422, 4 3 2 371293 k + 1283048 k + 1662804 k + 957852 k + 206933], [11 + 13 k, 3 + 3 k, -4 - 5 k, -4 - 4 k, 6 + 7 k]] 2 PISOT(13, 169 k + 147) = [[13, 169 k + 147, 2197 k + 3822 k + 1662, 3 2 28561 k + 74529 k + 64821 k + 18791, 4 3 2 371293 k + 1291836 k + 1685385 k + 977194 k + 212456], [11 + 13 k, 3 + 4 k, 5 + 6 k, 2 + 3 k, 7 + 8 k]] PISOT(13, 169 k + 148) = [[13, 169 k + 148], [12 + 13 k, -7 - 8 k]] PISOT(13, 169 k + 150) = [ 2 [13, 169 k + 150, 2197 k + 3900 k + 1731], [12 + 13 k, -6 - 6 k, 8 + 9 k]] PISOT(13, 169 k + 154) = [[13, 169 k + 154], [11 + 13 k, 10 + 11 k]] PISOT(13, 169 k + 155) = [[13, 169 k + 155], [11 + 13 k, 11 + 12 k]] 2 PISOT(13, 169 k + 161) = [[13, 169 k + 161, 2197 k + 4186 k + 1994, 3 2 28561 k + 81627 k + 77765 k + 24696, 4 3 2 371293 k + 1414868 k + 2021877 k + 1284164 k + 305864, 5 4 3 2 4826809 k + 22991605 k + 43807166 k + 41734805 k + 19880612 k + 3788176 ], [13 + 13 k, -8 - 8 k, 5 + 5 k, -3 - 3 k, 2 + 2 k, -1 - k]] 2 PISOT(13, 169 k + 162) = [[13, 169 k + 162, 2197 k + 4212 k + 2019, 3 2 28561 k + 82134 k + 78738 k + 25163, 4 3 2 371293 k + 1423656 k + 2047149 k + 1308394 k + 313609], [13 + 13 k, -7 - 7 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(13, 169 k + 163) = [[13, 169 k + 163, 2197 k + 4238 k + 2044, 3 2 28561 k + 82641 k + 79713 k + 25632], [13 + 13 k, -6 - 6 k, 3 + 3 k, -1 - k]] 2 PISOT(13, 169 k + 164) = [[13, 169 k + 164, 2197 k + 4264 k + 2069, 3 2 28561 k + 83148 k + 80690 k + 26102], [13 + 13 k, -5 - 5 k, 2 + 2 k, -1 - k]] PISOT(13, 169 k + 165) = 2 [[13, 169 k + 165, 2197 k + 4290 k + 2094], [13 + 13 k, -4 - 4 k, 1 + k]] PISOT(13, 169 k + 166) = 2 [[13, 169 k + 166, 2197 k + 4316 k + 2120], [13 + 13 k, -3 - 3 k, 1 + k]] PISOT(13, 169 k + 167) = [[13, 169 k + 167], [13 + 13 k, -2 - 2 k]] PISOT(13, 169 k + 168) = [[13, 169 k + 168], [13 + 13 k, -1 - k]] PISOT(14, 196 k + 1) = [[14, 196 k + 1], [14 k, k]] PISOT(14, 196 k + 2) = [[14, 196 k + 2], [14 k, 2 k]] 2 PISOT(14, 196 k + 3) = [[14, 196 k + 3, 2744 k + 84 k + 1], [14 k, 3 k, k]] 2 PISOT(14, 196 k + 4) = [[14, 196 k + 4, 2744 k + 112 k + 1], [14 k, 4 k, k]] PISOT(14, 196 k + 5) = [ 2 3 2 [14, 196 k + 5, 2744 k + 140 k + 2, 38416 k + 2940 k + 81 k + 1], [14 k, 5 k, 2 k, k]] PISOT(14, 196 k + 6) = [ 2 3 2 [14, 196 k + 6, 2744 k + 168 k + 3, 38416 k + 3528 k + 120 k + 1], [14 k, 6 k, 3 k, k]] 2 PISOT(14, 196 k + 7) = [[14, 196 k + 7, 2744 k + 196 k + 4, 3 2 4 3 2 38416 k + 4116 k + 161 k + 2, 537824 k + 76832 k + 4410 k + 112 k + 1] , [14 k, 7 k, 4 k, 2 k, k]] 2 PISOT(14, 196 k + 8) = [[14, 196 k + 8, 2744 k + 224 k + 5, 3 2 4 3 2 38416 k + 4704 k + 204 k + 3, 537824 k + 87808 k + 5628 k + 164 k + 2, 5 4 3 2 7529536 k + 1536640 k + 130144 k + 5636 k + 129 k + 1], [1 + 14 k, -6 k, -3 k, -2 k, -k, -k]] PISOT(14, 196 k + 15) = [[14, 196 k + 15], [2 + 14 k, -1 - 13 k]] PISOT(14, 196 k + 16) = [[14, 196 k + 16], [2 + 14 k, -1 - 12 k]] PISOT(14, 196 k + 19) = [ 2 3 2 [14, 196 k + 19, 2744 k + 532 k + 26, 38416 k + 11172 k + 1089 k + 36], [1 + 14 k, 5 k, 7 k, 1 + 10 k]] PISOT(14, 196 k + 29) = [ 2 3 2 [14, 196 k + 29, 2744 k + 812 k + 60, 38416 k + 17052 k + 2521 k + 124], [1 + 14 k, 1 + 15 k, 2 + 17 k, 1 + 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.991479277104490 2 PISOT(14, 196 k + 31) = [[14, 196 k + 31, 2744 k + 868 k + 69, 3 2 38416 k + 18228 k + 2893 k + 154, 4 3 2 537824 k + 340256 k + 80934 k + 8590 k + 344, 5 4 3 2 7529536 k + 5954480 k + 1887480 k + 300019 k + 23941 k + 768], [3 + 14 k, -2 - 11 k, 1 + 4 k, -1 - 5 k, 3 k, 1 + 6 k]] PISOT(14, 196 k + 33) = 2 [[14, 196 k + 33, 2744 k + 924 k + 78], [2 + 14 k, 5 k, 2 + 12 k]] PISOT(14, 196 k + 35) = 2 [[14, 196 k + 35, 2744 k + 980 k + 88], [1 + 14 k, 3 + 21 k, 2 + 11 k]] 2 PISOT(14, 196 k + 38) = [[14, 196 k + 38, 2744 k + 1064 k + 103, 3 2 38416 k + 22344 k + 4328 k + 279, 4 3 2 537824 k + 417088 k + 121212 k + 15640 k + 756], [2 + 14 k, 1 + 10 k, 2 + 13 k, 1 + 7 k, 1 + 5 k]] PISOT(14, 196 k + 39) = [[ 2 3 2 14, 196 k + 39, 2744 k + 1092 k + 109, 38416 k + 22932 k + 4573 k + 305] , [2 + 14 k, 2 + 11 k, 1 + 3 k, -1 - 5 k]] PISOT(14, 196 k + 40) = [[ 2 3 2 14, 196 k + 40, 2744 k + 1120 k + 114, 38416 k + 23520 k + 4792 k + 325] , [3 + 14 k, -1 - 2 k, 2 + 8 k, -1 - 5 k]] PISOT(14, 196 k + 43) = [[ 2 3 2 14, 196 k + 43, 2744 k + 1204 k + 132, 38416 k + 25284 k + 5545 k + 405] , [3 + 14 k, k, 3 k, 2 + 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.982980847864889 PISOT(14, 196 k + 44) = [[ 2 3 2 14, 196 k + 44, 2744 k + 1232 k + 138, 38416 k + 25872 k + 5800 k + 433] , [3 + 14 k, 2 k, 2 + 6 k, -2 - 9 k]] PISOT(14, 196 k + 45) = 2 [[14, 196 k + 45, 2744 k + 1260 k + 145], [2 + 14 k, 3 + 17 k, 3 + 13 k]] 2 PISOT(14, 196 k + 47) = [[14, 196 k + 47, 2744 k + 1316 k + 158, 3 2 38416 k + 27636 k + 6633 k + 531, 4 3 2 537824 k + 515872 k + 185682 k + 29720 k + 1785], [4 + 14 k, -2 - 9 k, -1 - 2 k, 2 + 7 k, -1 - 4 k]] PISOT(14, 196 k + 48) = [[ 2 3 2 14, 196 k + 48, 2744 k + 1344 k + 165, 38416 k + 28224 k + 6924 k + 567] , [3 + 14 k, 1 + 6 k, 2 + 7 k, -1 - 4 k]] PISOT(14, 196 k + 49) = 2 [[14, 196 k + 49, 2744 k + 1372 k + 172], [4 + 14 k, -2 - 7 k, 1 + 4 k]] PISOT(14, 196 k + 50) = [[ 2 3 2 14, 196 k + 50, 2744 k + 1400 k + 179, 38416 k + 29400 k + 7512 k + 641] , [3 + 14 k, 2 + 8 k, k, 1 + 4 k]] PISOT(14, 196 k + 51) = [[ 2 3 2 14, 196 k + 51, 2744 k + 1428 k + 186, 38416 k + 29988 k + 7809 k + 678] , [3 + 14 k, 2 + 9 k, 1 + 5 k, 1 + 4 k]] PISOT(14, 196 k + 53) = [[14, 196 k + 53], [3 + 14 k, 3 + 11 k]] PISOT(14, 196 k + 59) = 2 [[14, 196 k + 59, 2744 k + 1652 k + 249], [5 + 14 k, -4 - 11 k, 3 + 10 k]] PISOT(14, 196 k + 60) = 2 [[14, 196 k + 60, 2744 k + 1680 k + 257], [5 + 14 k, -4 - 10 k, 4 + 13 k]] PISOT(14, 196 k + 63) = 2 [[14, 196 k + 63, 2744 k + 1764 k + 284], [5 + 14 k, -2 - 7 k, -1 - 3 k]] PISOT(14, 196 k + 65) = [[14, 196 k + 65], [4 + 14 k, 3 + 9 k]] PISOT(14, 196 k + 66) = 2 [[14, 196 k + 66, 2744 k + 1848 k + 311], [5 + 14 k, -2 - 4 k, 3 + 9 k]] PISOT(14, 196 k + 67) = [[14, 196 k + 67], [5 + 14 k, -1 - 3 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998961284771711 2 PISOT(14, 196 k + 68) = [[14, 196 k + 68, 2744 k + 1904 k + 330, 3 2 38416 k + 39984 k + 13864 k + 1601, 4 3 2 537824 k + 746368 k + 388248 k + 89708 k + 7767], [4 + 14 k, 4 + 12 k, 1 + 2 k, -2 - 5 k, 1 + 3 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.981668422482757 2 PISOT(14, 196 k + 71) = [[14, 196 k + 71, 2744 k + 1988 k + 360, 3 2 38416 k + 41748 k + 15121 k + 1825], [5 + 14 k, k, 1 + 5 k, 4 + 11 k]] PISOT(14, 196 k + 75) = 2 [[14, 196 k + 75, 2744 k + 2100 k + 402], [5 + 14 k, 1 + 5 k, 5 + 13 k]] 2 PISOT(14, 196 k + 77) = [[14, 196 k + 77, 2744 k + 2156 k + 424, 3 2 38416 k + 45276 k + 17801 k + 2335], [5 + 14 k, 2 + 7 k, 4 + 11 k, 2 + 5 k]] PISOT(14, 196 k + 78) = 2 [[14, 196 k + 78, 2744 k + 2184 k + 435], [6 + 14 k, -2 - 6 k, -2 - 5 k]] PISOT(14, 196 k + 79) = [[14, 196 k + 79], [6 + 14 k, -2 - 5 k]] PISOT(14, 196 k + 82) = [[14, 196 k + 82], [5 + 14 k, 5 + 12 k]] 2 PISOT(14, 196 k + 83) = [[14, 196 k + 83, 2744 k + 2324 k + 492, 3 2 38416 k + 48804 k + 20665 k + 2916, 4 3 2 537824 k + 911008 k + 578634 k + 163320 k + 17283], [5 + 14 k, 5 + 13 k, 2 + 7 k, 5 + 13 k, 3 + 7 k]] PISOT(14, 196 k + 85) = 2 [[14, 196 k + 85, 2744 k + 2380 k + 516], [5 + 14 k, 6 + 15 k, 3 + 7 k]] PISOT(14, 196 k + 87) = 2 [[14, 196 k + 87, 2744 k + 2436 k + 541], [6 + 14 k, 2 + 3 k, -4 - 9 k]] 2 PISOT(14, 196 k + 88) = [[14, 196 k + 88, 2744 k + 2464 k + 553, 3 2 38416 k + 51744 k + 23228 k + 3475, 4 3 2 537824 k + 965888 k + 650412 k + 194628 k + 21837], [7 + 14 k, -5 - 10 k, 4 + 7 k, -6 - 12 k, 4 + 9 k]] PISOT(14, 196 k + 96) = [[14, 196 k + 96], [7 + 14 k, -1 - 2 k]] PISOT(14, 196 k + 97) = 2 [[14, 196 k + 97, 2744 k + 2716 k + 672], [6 + 14 k, 6 + 13 k, 3 + 6 k]] PISOT(14, 196 k + 99) = 2 [[14, 196 k + 99, 2744 k + 2772 k + 700], [8 + 14 k, -7 - 13 k, 3 + 6 k]] PISOT(14, 196 k + 100) = [[14, 196 k + 100], [7 + 14 k, 1 + 2 k]] 2 PISOT(14, 196 k + 108) = [[14, 196 k + 108, 2744 k + 3024 k + 833, 3 2 38416 k + 63504 k + 34988 k + 6425, 4 3 2 537824 k + 1185408 k + 979692 k + 359828 k + 49557], [7 + 14 k, 5 + 10 k, 3 + 7 k, 6 + 12 k, 5 + 9 k]] PISOT(14, 196 k + 109) = 2 [[14, 196 k + 109, 2744 k + 3052 k + 849], [8 + 14 k, -1 - 3 k, -5 - 9 k]] PISOT(14, 196 k + 111) = 2 [[14, 196 k + 111, 2744 k + 3108 k + 880], [9 + 14 k, -9 - 15 k, 4 + 7 k]] 2 PISOT(14, 196 k + 113) = [[14, 196 k + 113, 2744 k + 3164 k + 912, 3 2 38416 k + 66444 k + 38305 k + 7361, 4 3 2 537824 k + 1240288 k + 1072554 k + 412220 k + 59413], [9 + 14 k, -8 - 13 k, 5 + 7 k, -8 - 13 k, 4 + 7 k]] PISOT(14, 196 k + 114) = [[14, 196 k + 114], [9 + 14 k, -7 - 12 k]] PISOT(14, 196 k + 117) = [[14, 196 k + 117], [8 + 14 k, 3 + 5 k]] PISOT(14, 196 k + 118) = 2 [[14, 196 k + 118, 2744 k + 3304 k + 995], [8 + 14 k, 4 + 6 k, -3 - 5 k]] 2 PISOT(14, 196 k + 119) = [[14, 196 k + 119, 2744 k + 3332 k + 1012, 3 2 38416 k + 69972 k + 42497 k + 8606], [9 + 14 k, -5 - 7 k, 7 + 11 k, -3 - 5 k]] PISOT(14, 196 k + 121) = [ 2 [14, 196 k + 121, 2744 k + 3388 k + 1046], [9 + 14 k, -4 - 5 k, 8 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.981668315388927 2 PISOT(14, 196 k + 125) = [[14, 196 k + 125, 2744 k + 3500 k + 1116, 3 2 38416 k + 73500 k + 46873 k + 9964], [9 + 14 k, -1 - k, 4 + 5 k, -7 - 11 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998962669113346 2 PISOT(14, 196 k + 128) = [[14, 196 k + 128, 2744 k + 3584 k + 1170, 3 2 38416 k + 75264 k + 49144 k + 10695, 4 3 2 537824 k + 1404928 k + 1376088 k + 598980 k + 97763], [10 + 14 k, -8 - 12 k, 1 + 2 k, 3 + 5 k, 2 + 3 k]] PISOT(14, 196 k + 129) = [[14, 196 k + 129], [9 + 14 k, 2 + 3 k]] PISOT(14, 196 k + 130) = 2 [[14, 196 k + 130, 2744 k + 3640 k + 1207], [9 + 14 k, 2 + 4 k, 6 + 9 k]] PISOT(14, 196 k + 131) = [[14, 196 k + 131], [10 + 14 k, -6 - 9 k]] PISOT(14, 196 k + 133) = 2 [[14, 196 k + 133, 2744 k + 3724 k + 1264], [9 + 14 k, 5 + 7 k, -2 - 3 k]] PISOT(14, 196 k + 136) = [ 2 [14, 196 k + 136, 2744 k + 3808 k + 1321], [9 + 14 k, 6 + 10 k, 9 + 13 k]] PISOT(14, 196 k + 137) = [ 2 [14, 196 k + 137, 2744 k + 3836 k + 1341], [9 + 14 k, 7 + 11 k, 7 + 10 k]] PISOT(14, 196 k + 143) = [[14, 196 k + 143], [11 + 14 k, -8 - 11 k]] 2 PISOT(14, 196 k + 145) = [[14, 196 k + 145, 2744 k + 4060 k + 1502, 3 2 38416 k + 85260 k + 63081 k + 15559], [11 + 14 k, -7 - 9 k, 4 + 5 k, -3 - 4 k]] 2 PISOT(14, 196 k + 146) = [[14, 196 k + 146, 2744 k + 4088 k + 1523, 3 2 38416 k + 85848 k + 63960 k + 15887], [11 + 14 k, -6 - 8 k, 1 + k, -3 - 4 k]] PISOT(14, 196 k + 147) = 2 [[14, 196 k + 147, 2744 k + 4116 k + 1544], [10 + 14 k, 5 + 7 k, 3 + 4 k]] 2 PISOT(14, 196 k + 148) = [[14, 196 k + 148, 2744 k + 4144 k + 1565, 3 2 38416 k + 87024 k + 65724 k + 16549], [11 + 14 k, -5 - 6 k, 5 + 7 k, 3 + 4 k]] 2 PISOT(14, 196 k + 149) = [[14, 196 k + 149, 2744 k + 4172 k + 1586, 3 2 38416 k + 87612 k + 66609 k + 16882, 4 3 2 537824 k + 1635424 k + 1865010 k + 945324 k + 179699], [10 + 14 k, 7 + 9 k, -1 - 2 k, -5 - 7 k, -3 - 4 k]] 2 PISOT(14, 196 k + 151) = [[14, 196 k + 151, 2744 k + 4228 k + 1629], [12 + 14 k, -14 - 17 k, 10 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.982982091476034 2 PISOT(14, 196 k + 152) = [[14, 196 k + 152, 2744 k + 4256 k + 1650, 3 2 38416 k + 89376 k + 69304 k + 17911], [11 + 14 k, -2 - 2 k, 4 + 6 k, 7 + 9 k]] 2 PISOT(14, 196 k + 153) = [[14, 196 k + 153, 2744 k + 4284 k + 1672, 3 2 38416 k + 89964 k + 70225 k + 18272], [11 + 14 k, -1 - k, 3 + 3 k, -7 - 9 k]] 2 PISOT(14, 196 k + 156) = [[14, 196 k + 156, 2744 k + 4368 k + 1738, 3 2 38416 k + 91728 k + 73000 k + 19363], [11 + 14 k, 1 + 2 k, 6 + 8 k, 4 + 5 k]] 2 PISOT(14, 196 k + 157) = [[14, 196 k + 157, 2744 k + 4396 k + 1761, 3 2 38416 k + 92316 k + 73957 k + 19752], [12 + 14 k, -9 - 11 k, 2 + 3 k, 4 + 5 k]] 2 PISOT(14, 196 k + 158) = [[14, 196 k + 158, 2744 k + 4424 k + 1783, 3 2 38416 k + 92904 k + 74888 k + 20121, 4 3 2 537824 k + 1734208 k + 2096892 k + 1126816 k + 227064], [12 + 14 k, -9 - 10 k, 11 + 13 k, -6 - 7 k, 4 + 5 k]] 2 PISOT(14, 196 k + 161) = [[14, 196 k + 161, 2744 k + 4508 k + 1852], [13 + 14 k, -18 - 21 k, 9 + 11 k]] 2 PISOT(14, 196 k + 163) = [[14, 196 k + 163, 2744 k + 4564 k + 1898], [12 + 14 k, -5 - 5 k, 10 + 12 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.991479183467174 2 PISOT(14, 196 k + 165) = [[14, 196 k + 165, 2744 k + 4620 k + 1945, 3 2 38416 k + 97020 k + 81685 k + 22927, 4 3 2 537824 k + 1811040 k + 2287110 k + 1283806 k + 270256, 5 4 3 2 7529536 k + 31693200 k + 53364920 k + 44930901 k + 18916103 k + 3185690 ], [11 + 14 k, 9 + 11 k, 3 + 4 k, 4 + 5 k, 3 + 3 k, -5 - 6 k]] 2 PISOT(14, 196 k + 167) = [[14, 196 k + 167, 2744 k + 4676 k + 1992, 3 2 38416 k + 98196 k + 83665 k + 23761], [13 + 14 k, -14 - 15 k, 15 + 17 k, -6 - 7 k]] 2 PISOT(14, 196 k + 177) = [[14, 196 k + 177, 2744 k + 4956 k + 2238, 3 2 38416 k + 104076 k + 93993 k + 28297], [13 + 14 k, -5 - 5 k, 7 + 7 k, -9 - 10 k]] PISOT(14, 196 k + 180) = [[14, 196 k + 180], [12 + 14 k, 11 + 12 k]] PISOT(14, 196 k + 181) = [[14, 196 k + 181], [12 + 14 k, 12 + 13 k]] 2 PISOT(14, 196 k + 188) = [[14, 196 k + 188, 2744 k + 5264 k + 2525, 3 2 38416 k + 110544 k + 106044 k + 33913, 4 3 2 537824 k + 2063488 k + 2969148 k + 1898964 k + 455482, 5 4 3 2 7529536 k + 36111040 k + 69278944 k + 66460316 k + 31880409 k + 6117532 ], [13 + 14 k, 6 + 6 k, -3 - 3 k, 2 + 2 k, -1 - k, 1 + k]] 2 PISOT(14, 196 k + 189) = [[14, 196 k + 189, 2744 k + 5292 k + 2552, 3 2 38416 k + 111132 k + 107177 k + 34459, 4 3 2 537824 k + 2074464 k + 3000858 k + 1929508 k + 465291], [14 + 14 k, -7 - 7 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(14, 196 k + 190) = [[14, 196 k + 190, 2744 k + 5320 k + 2579, 3 2 38416 k + 111720 k + 108312 k + 35007], [14 + 14 k, -6 - 6 k, 3 + 3 k, -1 - k]] 2 PISOT(14, 196 k + 191) = [[14, 196 k + 191, 2744 k + 5348 k + 2606, 3 2 38416 k + 112308 k + 109449 k + 35556], [14 + 14 k, -5 - 5 k, 2 + 2 k, -1 - k]] PISOT(14, 196 k + 192) = 2 [[14, 196 k + 192, 2744 k + 5376 k + 2633], [14 + 14 k, -4 - 4 k, 1 + k]] PISOT(14, 196 k + 193) = 2 [[14, 196 k + 193, 2744 k + 5404 k + 2661], [14 + 14 k, -3 - 3 k, 1 + k]] PISOT(14, 196 k + 194) = [[14, 196 k + 194], [14 + 14 k, -2 - 2 k]] PISOT(14, 196 k + 195) = [[14, 196 k + 195], [14 + 14 k, -1 - k]] PISOT(15, 225 k + 1) = [[15, 225 k + 1], [15 k, k]] PISOT(15, 225 k + 2) = [[15, 225 k + 2], [15 k, 2 k]] 2 PISOT(15, 225 k + 3) = [[15, 225 k + 3, 3375 k + 90 k + 1], [15 k, 3 k, k]] 2 PISOT(15, 225 k + 4) = [[15, 225 k + 4, 3375 k + 120 k + 1], [15 k, 4 k, k]] PISOT(15, 225 k + 5) = [ 2 3 2 [15, 225 k + 5, 3375 k + 150 k + 2, 50625 k + 3375 k + 85 k + 1], [15 k, 5 k, 2 k, k]] PISOT(15, 225 k + 6) = [ 2 3 2 [15, 225 k + 6, 3375 k + 180 k + 2, 50625 k + 4050 k + 96 k + 1], [15 k, 6 k, 2 k, k]] PISOT(15, 225 k + 7) = [ 2 3 2 [15, 225 k + 7, 3375 k + 210 k + 3, 50625 k + 4725 k + 139 k + 1], [15 k, 7 k, 3 k, k]] 2 PISOT(15, 225 k + 8) = [[15, 225 k + 8, 3375 k + 240 k + 4, 3 2 4 3 2 50625 k + 5400 k + 184 k + 2, 759375 k + 108000 k + 5580 k + 124 k + 1 ], [15 k, 8 k, 4 k, 2 k, k]] 2 PISOT(15, 225 k + 9) = [[15, 225 k + 9, 3375 k + 270 k + 5, 3 2 4 3 2 50625 k + 6075 k + 231 k + 3, 759375 k + 121500 k + 7020 k + 180 k + 2, 5 4 3 2 11390625 k + 2278125 k + 176850 k + 6804 k + 139 k + 1], [15 k, 9 k, 5 k, 3 k, 2 k, k]] PISOT(15, 225 k + 16) = [[15, 225 k + 16], [2 + 15 k, -1 - 14 k]] PISOT(15, 225 k + 17) = [[15, 225 k + 17], [2 + 15 k, -1 - 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.978222659277365 PISOT(15, 225 k + 20) = [ 2 3 2 [15, 225 k + 20, 3375 k + 600 k + 27, 50625 k + 13500 k + 1210 k + 36], [1 + 15 k, 1 + 5 k, -8 k, -1 - 11 k]] PISOT(15, 225 k + 21) = [ 2 3 2 [15, 225 k + 21, 3375 k + 630 k + 29, 50625 k + 14175 k + 1311 k + 40], [1 + 15 k, 6 k, 8 k, 1 + 11 k]] PISOT(15, 225 k + 22) = 2 [[15, 225 k + 22, 3375 k + 660 k + 32], [1 + 15 k, 7 k, 1 + 10 k]] PISOT(15, 225 k + 28) = [ 2 3 2 [15, 225 k + 28, 3375 k + 840 k + 52, 50625 k + 18900 k + 2344 k + 97], [2 + 15 k, -2 k, -1 - 4 k, 1 + 8 k]] PISOT(15, 225 k + 29) = [ 2 3 2 [15, 225 k + 29, 3375 k + 870 k + 56, 50625 k + 19575 k + 2521 k + 108], [1 + 15 k, 1 + 14 k, 1 + 12 k, 1 + 8 k]] 2 PISOT(15, 225 k + 31) = [[15, 225 k + 31, 3375 k + 930 k + 64, 3 2 50625 k + 20925 k + 2881 k + 132, 4 3 2 759375 k + 418500 k + 86445 k + 7928 k + 272], [2 + 15 k, k, 2 k, 1 + 4 k, -1 - 7 k]] PISOT(15, 225 k + 32) = [ 2 3 2 [15, 225 k + 32, 3375 k + 960 k + 68, 50625 k + 21600 k + 3064 k + 144], [2 + 15 k, 2 k, 1 + 4 k, -1 - 7 k]] PISOT(15, 225 k + 33) = 2 [[15, 225 k + 33, 3375 k + 990 k + 73], [2 + 15 k, 3 k, 1 + 7 k]] 2 PISOT(15, 225 k + 38) = [[15, 225 k + 38, 3375 k + 1140 k + 96, 3 2 50625 k + 25650 k + 4324 k + 243, 4 3 2 759375 k + 513000 k + 129780 k + 14586 k + 615], [3 + 15 k, -1 - 7 k, -3 k, -2 - 7 k, 2 + 12 k]] PISOT(15, 225 k + 41) = [[15, 225 k + 41], [2 + 15 k, 2 + 11 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.973283997673568 PISOT(15, 225 k + 44) = [[ 2 3 2 15, 225 k + 44, 3375 k + 1320 k + 129, 50625 k + 29700 k + 5806 k + 378] , [3 + 15 k, -1 - k, 2 + 12 k, 1 + 5 k]] 2 PISOT(15, 225 k + 54) = [[15, 225 k + 54, 3375 k + 1620 k + 194, 3 2 50625 k + 36450 k + 8736 k + 697, 4 3 2 759375 k + 729000 k + 262170 k + 41862 k + 2504, 5 4 3 2 11390625 k + 13668750 k + 6555600 k + 1570779 k + 188032 k + 8996], [4 + 15 k, -1 - 6 k, -2 - 7 k, 1 + 5 k, 1 + 3 k, -1 - 4 k]] PISOT(15, 225 k + 55) = [[ 2 3 2 15, 225 k + 55, 3375 k + 1650 k + 202, 50625 k + 37125 k + 9085 k + 742] , [4 + 15 k, -1 - 5 k, -1 - 3 k, 1 + 4 k]] PISOT(15, 225 k + 56) = [[15, 225 k + 56], [4 + 15 k, -1 - 4 k]] PISOT(15, 225 k + 57) = 2 [[15, 225 k + 57, 3375 k + 1710 k + 217], [4 + 15 k, -1 - 3 k, 1 + 4 k]] PISOT(15, 225 k + 58) = [[ 2 3 2 15, 225 k + 58, 3375 k + 1740 k + 224, 50625 k + 39150 k + 10084 k + 865 ], [3 + 15 k, 3 + 13 k, 1 + 5 k, 1 + 4 k]] PISOT(15, 225 k + 61) = 2 [[15, 225 k + 61, 3375 k + 1830 k + 248], [4 + 15 k, 1 + k, -3 - 11 k]] 2 PISOT(15, 225 k + 64) = [[15, 225 k + 64, 3375 k + 1920 k + 273, 3 2 50625 k + 43200 k + 12286 k + 1165], [5 + 15 k, -3 - 11 k, -1 - 2 k, 2 + 7 k]] PISOT(15, 225 k + 65) = 2 [[15, 225 k + 65, 3375 k + 1950 k + 282], [4 + 15 k, 1 + 5 k, 2 + 7 k]] 2 PISOT(15, 225 k + 72) = [[15, 225 k + 72, 3375 k + 2160 k + 346, 3 2 50625 k + 48600 k + 15564 k + 1663], [4 + 15 k, 3 + 12 k, 4 + 13 k, 1 + 3 k]] 2 PISOT(15, 225 k + 73) = [[15, 225 k + 73, 3375 k + 2190 k + 355, 3 2 50625 k + 49275 k + 15979 k + 1726, 4 3 2 759375 k + 985500 k + 479430 k + 103610 k + 8392, 5 4 3 2 11390625 k + 18478125 k + 11986650 k + 3886417 k + 629781 k + 40803], [4 + 15 k, 3 + 13 k, 5 + 18 k, 3 + 12 k, 4 + 13 k, 1 + 3 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.971994603480665 2 PISOT(15, 225 k + 74) = [[15, 225 k + 74, 3375 k + 2220 k + 365, 3 2 50625 k + 49950 k + 16426 k + 1800, 4 3 2 759375 k + 999000 k + 492795 k + 108020 k + 8877, 5 4 3 2 11390625 k + 18731250 k + 12320100 k + 4051124 k + 665935 k + 43778], [5 + 15 k, -1 - k, 3 + 10 k, 1 + 4 k, 1 + 5 k, 3 + 9 k]] PISOT(15, 225 k + 76) = 2 [[15, 225 k + 76, 3375 k + 2280 k + 385], [4 + 15 k, 5 + 16 k, 2 + 6 k]] 2 PISOT(15, 225 k + 78) = [[15, 225 k + 78, 3375 k + 2340 k + 406, 3 2 50625 k + 52650 k + 18264 k + 2113, 4 3 2 759375 k + 1053000 k + 547830 k + 126726 k + 10997, 5 4 3 2 11390625 k + 19743750 k + 13694400 k + 4750947 k + 824374 k + 57233], [5 + 15 k, 1 + 3 k, k, 1 + 5 k, 4 + 11 k, -1 - 3 k]] 2 PISOT(15, 225 k + 82) = [[15, 225 k + 82, 3375 k + 2460 k + 448, 3 2 50625 k + 55350 k + 20164 k + 2448, 4 3 2 759375 k + 1107000 k + 604980 k + 146912 k + 13377], [6 + 15 k, -3 - 8 k, 1 + k, -4 - 9 k, 4 + 11 k]] 2 PISOT(15, 225 k + 84) = [[15, 225 k + 84, 3375 k + 2520 k + 470, 3 2 50625 k + 56700 k + 21156 k + 2630], [6 + 15 k, -2 - 6 k, -2 - 4 k, 3 + 8 k]] 2 PISOT(15, 225 k + 85) = [[15, 225 k + 85, 3375 k + 2550 k + 482, 3 2 50625 k + 57375 k + 21685 k + 2733], [5 + 15 k, 3 + 10 k, 4 + 12 k, 3 + 8 k]] PISOT(15, 225 k + 87) = 2 [[15, 225 k + 87, 3375 k + 2610 k + 505], [6 + 15 k, -2 - 3 k, 5 + 13 k]] PISOT(15, 225 k + 91) = 2 [[15, 225 k + 91, 3375 k + 2730 k + 552], [7 + 15 k, -6 - 14 k, 2 + 5 k]] PISOT(15, 225 k + 95) = 2 [[15, 225 k + 95, 3375 k + 2850 k + 602], [5 + 15 k, 8 + 20 k, 3 + 7 k]] PISOT(15, 225 k + 96) = 2 [[15, 225 k + 96, 3375 k + 2880 k + 614], [6 + 15 k, 3 + 6 k, -3 - 7 k]] PISOT(15, 225 k + 97) = [[15, 225 k + 97], [6 + 15 k, 3 + 7 k]] 2 PISOT(15, 225 k + 98) = [[15, 225 k + 98, 3375 k + 2940 k + 640, 3 2 50625 k + 66150 k + 28804 k + 4180], [6 + 15 k, 4 + 8 k, -3 - 8 k, -3 - 7 k]] 2 PISOT(15, 225 k + 99) = [[15, 225 k + 99, 3375 k + 2970 k + 653, 3 2 50625 k + 66825 k + 29391 k + 4307], [7 + 15 k, -2 - 6 k, -5 - 10 k, 4 + 9 k]] 2 PISOT(15, 225 k + 102) = [[15, 225 k + 102, 3375 k + 3060 k + 694, 3 2 50625 k + 68850 k + 31224 k + 4722], [7 + 15 k, -1 - 3 k, -3 - 5 k, 5 + 11 k]] 2 PISOT(15, 225 k + 109) = [[15, 225 k + 109, 3375 k + 3270 k + 792, 3 2 50625 k + 73575 k + 35641 k + 5755], [8 + 15 k, -6 - 11 k, 5 + 10 k, -1 - 2 k]] PISOT(15, 225 k + 112) = [[15, 225 k + 112], [8 + 15 k, -4 - 8 k]] PISOT(15, 225 k + 113) = [[15, 225 k + 113], [7 + 15 k, 4 + 8 k]] 2 PISOT(15, 225 k + 116) = [[15, 225 k + 116, 3375 k + 3480 k + 897, 3 2 50625 k + 78300 k + 40366 k + 6936], [7 + 15 k, 5 + 11 k, 5 + 10 k, 1 + 2 k]] 2 PISOT(15, 225 k + 123) = [[15, 225 k + 123, 3375 k + 3690 k + 1009, 3 2 50625 k + 83025 k + 45399 k + 8277], [8 + 15 k, 2 + 3 k, -2 - 5 k, -6 - 11 k]] 2 PISOT(15, 225 k + 126) = [[15, 225 k + 126, 3375 k + 3780 k + 1058, 3 2 50625 k + 85050 k + 47616 k + 8884], [8 + 15 k, 4 + 6 k, -5 - 10 k, -5 - 9 k]] 2 PISOT(15, 225 k + 127) = [[15, 225 k + 127, 3375 k + 3810 k + 1075, 3 2 50625 k + 85725 k + 48379 k + 9099], [9 + 15 k, -4 - 8 k, -5 - 8 k, 4 + 7 k]] PISOT(15, 225 k + 128) = [[15, 225 k + 128], [9 + 15 k, -4 - 7 k]] PISOT(15, 225 k + 129) = [ 2 [15, 225 k + 129, 3375 k + 3870 k + 1109], [9 + 15 k, -3 - 6 k, -4 - 7 k]] 2 PISOT(15, 225 k + 130) = [[15, 225 k + 130, 3375 k + 3900 k + 1127], [10 + 15 k, -12 - 20 k, 4 + 7 k]] PISOT(15, 225 k + 134) = 2 [[15, 225 k + 134, 3375 k + 4020 k + 1197], [8 + 15 k, 8 + 14 k, 3 + 5 k]] PISOT(15, 225 k + 138) = 2 [[15, 225 k + 138, 3375 k + 4140 k + 1270], [9 + 15 k, 1 + 3 k, 8 + 13 k]] 2 PISOT(15, 225 k + 140) = [[15, 225 k + 140, 3375 k + 4200 k + 1307, 3 2 50625 k + 94500 k + 58810 k + 12202], [10 + 15 k, -7 - 10 k, 8 + 12 k, -5 - 8 k]] 2 PISOT(15, 225 k + 141) = [[15, 225 k + 141, 3375 k + 4230 k + 1325, 3 2 50625 k + 95175 k + 59631 k + 12451], [9 + 15 k, 4 + 6 k, -2 - 4 k, -5 - 8 k]] 2 PISOT(15, 225 k + 143) = [[15, 225 k + 143, 3375 k + 4290 k + 1363, 3 2 50625 k + 96525 k + 61339 k + 12991, 4 3 2 759375 k + 1930500 k + 1840230 k + 779548 k + 123820], [9 + 15 k, 5 + 8 k, k, 5 + 9 k, 7 + 11 k]] 2 PISOT(15, 225 k + 147) = [[15, 225 k + 147, 3375 k + 4410 k + 1441, 3 2 50625 k + 99225 k + 64839 k + 14126, 4 3 2 5 759375 k + 1984500 k + 1945080 k + 847434 k + 138476, 11390625 k 4 3 2 + 37209375 k + 48625650 k + 31776003 k + 10383805 k + 1357469], [10 + 15 k, -2 - 3 k, 1 + k, -4 - 5 k, 7 + 11 k, 2 + 3 k]] 2 PISOT(15, 225 k + 149) = [[15, 225 k + 149, 3375 k + 4470 k + 1480], [11 + 15 k, -11 - 16 k, 4 + 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.971994371934685 2 PISOT(15, 225 k + 151) = [[15, 225 k + 151, 3375 k + 4530 k + 1520, 3 2 50625 k + 101925 k + 68401 k + 15301, 4 3 2 5 759375 k + 2038500 k + 2052045 k + 918070 k + 154027, 11390625 k 4 3 2 + 38221875 k + 51301350 k + 34427926 k + 11552112 k + 1550508], [10 + 15 k, k, 7 + 10 k, -3 - 4 k, 4 + 5 k, -6 - 9 k]] 2 PISOT(15, 225 k + 152) = [[15, 225 k + 152, 3375 k + 4560 k + 1540, 3 2 50625 k + 102600 k + 69304 k + 15603, 4 3 2 5 759375 k + 2052000 k + 2079180 k + 936250 k + 158087, 11390625 k 4 3 2 + 38475000 k + 51980400 k + 35111033 k + 11857522 k + 1601711], [11 + 15 k, -10 - 13 k, 13 + 18 k, -9 - 12 k, 9 + 13 k, -2 - 3 k]] 2 PISOT(15, 225 k + 153) = [[15, 225 k + 153, 3375 k + 4590 k + 1561, 3 2 50625 k + 103275 k + 70239 k + 15926], [11 + 15 k, -9 - 12 k, 9 + 13 k, -2 - 3 k]] PISOT(15, 225 k + 160) = [ 2 [15, 225 k + 160, 3375 k + 4800 k + 1707], [11 + 15 k, -4 - 5 k, 5 + 7 k]] 2 PISOT(15, 225 k + 161) = [[15, 225 k + 161, 3375 k + 4830 k + 1728, 3 2 50625 k + 108675 k + 77761 k + 18546], [10 + 15 k, 8 + 11 k, -1 - 2 k, -5 - 7 k]] PISOT(15, 225 k + 164) = 2 [[15, 225 k + 164, 3375 k + 4920 k + 1793], [11 + 15 k, -k, -8 - 11 k]] 2 PISOT(15, 225 k + 167) = [[15, 225 k + 167, 3375 k + 5010 k + 1859, 3 2 50625 k + 112725 k + 83659 k + 20694], [12 + 15 k, -10 - 13 k, 4 + 5 k, -3 - 4 k]] PISOT(15, 225 k + 168) = 2 [[15, 225 k + 168, 3375 k + 5040 k + 1882], [11 + 15 k, 2 + 3 k, 3 + 4 k]] PISOT(15, 225 k + 169) = [[15, 225 k + 169], [11 + 15 k, 3 + 4 k]] 2 PISOT(15, 225 k + 170) = [[15, 225 k + 170, 3375 k + 5100 k + 1927, 3 2 50625 k + 114750 k + 86710 k + 21843], [11 + 15 k, 4 + 5 k, -2 - 3 k, -3 - 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.973283662109210 2 PISOT(15, 225 k + 181) = [[15, 225 k + 181, 3375 k + 5430 k + 2184, 3 2 50625 k + 122175 k + 98281 k + 26353], [12 + 15 k, k, 10 + 12 k, -4 - 5 k]] PISOT(15, 225 k + 184) = [[15, 225 k + 184], [13 + 15 k, -9 - 11 k]] 2 PISOT(15, 225 k + 187) = [[15, 225 k + 187, 3375 k + 5610 k + 2331, 3 2 50625 k + 126225 k + 104899 k + 29056, 4 3 2 759375 k + 2524500 k + 3147030 k + 1743474 k + 362184], [12 + 15 k, 6 + 7 k, -3 - 3 k, 5 + 7 k, 10 + 12 k]] PISOT(15, 225 k + 192) = [ 2 [15, 225 k + 192, 3375 k + 5760 k + 2458], [13 + 15 k, -3 - 3 k, 6 + 7 k]] 2 PISOT(15, 225 k + 193) = [[15, 225 k + 193, 3375 k + 5790 k + 2483, 3 2 50625 k + 130275 k + 111739 k + 31945], [13 + 15 k, -2 - 2 k, 3 + 4 k, 6 + 7 k]] 2 PISOT(15, 225 k + 194) = [[15, 225 k + 194, 3375 k + 5820 k + 2509, 3 2 50625 k + 130950 k + 112906 k + 32449, 4 3 2 759375 k + 2619000 k + 3387195 k + 1946962 k + 419664], [13 + 15 k, -1 - k, 2 + 2 k, -3 - 4 k, -6 - 7 k]] 2 PISOT(15, 225 k + 196) = [[15, 225 k + 196, 3375 k + 5880 k + 2561, 3 2 50625 k + 132300 k + 115246 k + 33463], [14 + 15 k, -13 - 14 k, 11 + 12 k, -7 - 8 k]] 2 PISOT(15, 225 k + 197) = [[15, 225 k + 197, 3375 k + 5910 k + 2587, 3 2 50625 k + 132975 k + 116419 k + 33972], [13 + 15 k, 2 + 2 k, -3 - 4 k, -7 - 8 k]] PISOT(15, 225 k + 203) = [ 2 [15, 225 k + 203, 3375 k + 6090 k + 2747], [14 + 15 k, -7 - 7 k, 9 + 10 k] ] 2 PISOT(15, 225 k + 204) = [[15, 225 k + 204, 3375 k + 6120 k + 2774, 3 2 50625 k + 137700 k + 124836 k + 37721], [14 + 15 k, -6 - 6 k, 8 + 8 k, -10 - 11 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.978221647569147 2 PISOT(15, 225 k + 205) = [[15, 225 k + 205, 3375 k + 6150 k + 2802, 3 2 50625 k + 138375 k + 126085 k + 38299], [14 + 15 k, -4 - 5 k, -8 - 8 k, 10 + 11 k]] PISOT(15, 225 k + 208) = [[15, 225 k + 208], [13 + 15 k, 12 + 13 k]] PISOT(15, 225 k + 209) = [[15, 225 k + 209], [13 + 15 k, 13 + 14 k]] 2 PISOT(15, 225 k + 216) = [[15, 225 k + 216, 3375 k + 6480 k + 3110, 3 2 50625 k + 145800 k + 139956 k + 44778, 4 3 2 5 759375 k + 2916000 k + 4198770 k + 2686860 k + 644717, 11390625 k 4 3 2 + 54675000 k + 104970600 k + 100761246 k + 48357706 k + 9282684], [15 + 15 k, -9 - 9 k, 5 + 5 k, -3 - 3 k, 2 + 2 k, -1 - k]] 2 PISOT(15, 225 k + 217) = [[15, 225 k + 217, 3375 k + 6510 k + 3139, 3 2 50625 k + 146475 k + 141259 k + 45407, 4 3 2 759375 k + 2929500 k + 4237830 k + 2724536 k + 656832], [15 + 15 k, -8 - 8 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(15, 225 k + 218) = [[15, 225 k + 218, 3375 k + 6540 k + 3168, 3 2 50625 k + 147150 k + 142564 k + 46038], [15 + 15 k, -7 - 7 k, 3 + 3 k, -1 - k]] 2 PISOT(15, 225 k + 219) = [[15, 225 k + 219, 3375 k + 6570 k + 3197, 3 2 50625 k + 147825 k + 143871 k + 46670], [15 + 15 k, -6 - 6 k, 2 + 2 k, -1 - k]] 2 PISOT(15, 225 k + 220) = [[15, 225 k + 220, 3375 k + 6600 k + 3227, 3 2 50625 k + 148500 k + 145210 k + 47334], [15 + 15 k, -5 - 5 k, 2 + 2 k, -1 - k]] PISOT(15, 225 k + 221) = 2 [[15, 225 k + 221, 3375 k + 6630 k + 3256], [15 + 15 k, -4 - 4 k, 1 + k]] PISOT(15, 225 k + 222) = 2 [[15, 225 k + 222, 3375 k + 6660 k + 3286], [15 + 15 k, -3 - 3 k, 1 + k]] PISOT(15, 225 k + 223) = [[15, 225 k + 223], [15 + 15 k, -2 - 2 k]] PISOT(15, 225 k + 224) = [[15, 225 k + 224], [15 + 15 k, -1 - k]] PISOT(16, 256 k + 1) = [[16, 256 k + 1], [16 k, k]] PISOT(16, 256 k + 2) = [[16, 256 k + 2], [16 k, 2 k]] 2 PISOT(16, 256 k + 3) = [[16, 256 k + 3, 4096 k + 96 k + 1], [16 k, 3 k, k]] 2 PISOT(16, 256 k + 4) = [[16, 256 k + 4, 4096 k + 128 k + 1], [16 k, 4 k, k]] PISOT(16, 256 k + 5) = [ 2 3 2 [16, 256 k + 5, 4096 k + 160 k + 2, 65536 k + 3840 k + 89 k + 1], [16 k, 5 k, 2 k, k]] PISOT(16, 256 k + 6) = [ 2 3 2 [16, 256 k + 6, 4096 k + 192 k + 2, 65536 k + 4608 k + 100 k + 1], [16 k, 6 k, 2 k, k]] PISOT(16, 256 k + 7) = [ 2 3 2 [16, 256 k + 7, 4096 k + 224 k + 3, 65536 k + 5376 k + 145 k + 1], [16 k, 7 k, 3 k, k]] 2 PISOT(16, 256 k + 8) = [[16, 256 k + 8, 4096 k + 256 k + 4, 3 2 65536 k + 6144 k + 192 k + 2, 4 3 2 1048576 k + 131072 k + 6144 k + 128 k + 1], [1 + 16 k, -8 k, -4 k, -2 k, -k]] 2 PISOT(16, 256 k + 9) = [[16, 256 k + 9, 4096 k + 288 k + 5, 3 2 65536 k + 6912 k + 241 k + 3, 4 3 2 1048576 k + 147456 k + 7728 k + 186 k + 2, 5 4 3 2 16777216 k + 2949120 k + 206336 k + 7353 k + 143 k + 1], [16 k, 9 k, 5 k, 3 k, 2 k, k]] PISOT(16, 256 k + 17) = [[16, 256 k + 17], [2 + 16 k, -1 - 15 k]] PISOT(16, 256 k + 18) = [[16, 256 k + 18], [2 + 16 k, -1 - 14 k]] PISOT(16, 256 k + 21) = 2 [[16, 256 k + 21, 4096 k + 672 k + 28], [16 k, 1 + 21 k, 1 + 12 k]] 2 PISOT(16, 256 k + 25) = [[16, 256 k + 25, 4096 k + 800 k + 39, 3 2 65536 k + 19200 k + 1873 k + 61, 4 3 2 1048576 k + 409600 k + 59952 k + 3902 k + 95, 5 4 3 2 16777216 k + 8192000 k + 1598976 k + 156073 k + 7611 k + 148], [2 + 16 k, -1 - 7 k, 5 k, 1 + 8 k, -1 - 4 k, 1 + 10 k]] PISOT(16, 256 k + 26) = [[16, 256 k + 26], [1 + 16 k, 1 + 10 k]] PISOT(16, 256 k + 28) = 2 [[16, 256 k + 28, 4096 k + 896 k + 49], [2 + 16 k, -1 - 4 k, 1 + 9 k]] 2 PISOT(16, 256 k + 31) = [[16, 256 k + 31, 4096 k + 992 k + 60, 3 2 65536 k + 23808 k + 2881 k + 116, 4 3 2 1048576 k + 507904 k + 92208 k + 7432 k + 224], [2 + 16 k, -k, -2 k, -1 - 4 k, 1 + 8 k]] 2 PISOT(16, 256 k + 33) = [[16, 256 k + 33, 4096 k + 1056 k + 68, 3 2 65536 k + 25344 k + 3265 k + 140, 4 3 2 1048576 k + 540672 k + 104496 k + 8968 k + 288], [2 + 16 k, k, 2 k, 4 k, 1 + 8 k]] 2 PISOT(16, 256 k + 35) = [[16, 256 k + 35, 4096 k + 1120 k + 77, 3 2 65536 k + 26880 k + 3689 k + 169, 4 3 2 1048576 k + 573440 k + 117936 k + 10798 k + 371, 5 4 3 2 16777216 k + 11468800 k + 3143168 k + 431387 k + 29631 k + 814], [1 + 16 k, 2 + 19 k, 1 + 10 k, 1 + 6 k, -3 k, -1 - 7 k]] PISOT(16, 256 k + 36) = 2 [[16, 256 k + 36, 4096 k + 1152 k + 81], [2 + 16 k, 1 + 4 k, -1 - 7 k]] PISOT(16, 256 k + 37) = 2 [[16, 256 k + 37, 4096 k + 1184 k + 86], [3 + 16 k, -2 - 11 k, 1 + 7 k]] PISOT(16, 256 k + 42) = [[16, 256 k + 42], [3 + 16 k, -1 - 6 k]] 2 PISOT(16, 256 k + 44) = [[16, 256 k + 44, 4096 k + 1408 k + 121, 3 2 65536 k + 33792 k + 5808 k + 333, 4 3 2 1048576 k + 720896 k + 185856 k + 21304 k + 916], [3 + 16 k, -1 - 4 k, 1 + 5 k, -2 k, -1 - 6 k]] PISOT(16, 256 k + 51) = 2 [[16, 256 k + 51, 4096 k + 1632 k + 163], [3 + 16 k, 3 k, 2 + 10 k]] 2 PISOT(16, 256 k + 56) = [[16, 256 k + 56, 4096 k + 1792 k + 196, 3 2 65536 k + 43008 k + 9408 k + 686, 4 3 2 1048576 k + 917504 k + 301056 k + 43904 k + 2401], [4 + 16 k, -2 - 8 k, 1 + 4 k, -1 - 2 k, 2 + 9 k]] PISOT(16, 256 k + 57) = [[16, 256 k + 57], [3 + 16 k, 2 + 9 k]] 2 PISOT(16, 256 k + 58) = [[16, 256 k + 58, 4096 k + 1856 k + 210, 3 2 65536 k + 44544 k + 10084 k + 760, 4 3 2 1048576 k + 950272 k + 322752 k + 48680 k + 2750, 5 4 3 2 16777216 k + 19005440 k + 8607744 k + 1948072 k + 220260 k + 9951], [4 + 16 k, -2 - 6 k, 2 + 10 k, 1 + 4 k, -1 - 2 k, 2 + 9 k]] 2 PISOT(16, 256 k + 62) = [[16, 256 k + 62, 4096 k + 1984 k + 240, 3 2 65536 k + 47616 k + 11524 k + 929, 4 3 2 1048576 k + 1015808 k + 368832 k + 59488 k + 3596], [4 + 16 k, -2 k, -2 - 8 k, k, 1 + 4 k]] PISOT(16, 256 k + 63) = 2 [[16, 256 k + 63, 4096 k + 2016 k + 248], [4 + 16 k, -k, -1 - 4 k]] PISOT(16, 256 k + 65) = 2 [[16, 256 k + 65, 4096 k + 2080 k + 264], [4 + 16 k, k, 1 + 4 k]] 2 PISOT(16, 256 k + 66) = [[16, 256 k + 66, 4096 k + 2112 k + 272, 3 2 65536 k + 50688 k + 13060 k + 1121, 4 3 2 1048576 k + 1081344 k + 417984 k + 71776 k + 4620], [4 + 16 k, 2 k, 2 + 8 k, k, 1 + 4 k]] PISOT(16, 256 k + 69) = 2 [[16, 256 k + 69, 4096 k + 2208 k + 298], [3 + 16 k, 5 + 21 k, 3 + 11 k]] 2 PISOT(16, 256 k + 70) = [[16, 256 k + 70, 4096 k + 2240 k + 306, 3 2 65536 k + 53760 k + 14692 k + 1338, 4 3 2 1048576 k + 1146880 k + 470208 k + 85656 k + 5850, 5 4 3 2 16777216 k + 22937600 k + 12539904 k + 3426904 k + 468156 k + 25577], [4 + 16 k, 2 + 6 k, -2 - 6 k, 2 + 6 k, -1 - 6 k, -3 - 11 k]] 2 PISOT(16, 256 k + 72) = [[16, 256 k + 72, 4096 k + 2304 k + 324, 3 2 65536 k + 55296 k + 15552 k + 1458, 4 3 2 1048576 k + 1179648 k + 497664 k + 93312 k + 6561], [5 + 16 k, -2 - 8 k, -1 - 4 k, -1 - 2 k, 2 + 7 k]] PISOT(16, 256 k + 73) = [[16, 256 k + 73], [5 + 16 k, -2 - 7 k]] 2 PISOT(16, 256 k + 74) = [[16, 256 k + 74, 4096 k + 2368 k + 342, 3 2 65536 k + 56832 k + 16420 k + 1581, 4 3 2 1048576 k + 1212416 k + 525504 k + 101208 k + 7309, 5 4 3 2 16777216 k + 24248320 k + 14014464 k + 4049000 k + 584840 k + 33790], [5 + 16 k, -2 - 6 k, 1 + 4 k, 1 + 3 k, -1 - 2 k, 2 + 7 k]] 2 PISOT(16, 256 k + 77) = [[16, 256 k + 77, 4096 k + 2464 k + 371, 3 2 65536 k + 59136 k + 17801 k + 1788], [5 + 16 k, -1 - 3 k, 2 k, 3 + 10 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998476198215330 2 PISOT(16, 256 k + 82) = [[16, 256 k + 82, 4096 k + 2624 k + 420, 3 2 65536 k + 62976 k + 20164 k + 2151], [4 + 16 k, 5 + 18 k, 4 + 12 k, -1 - 3 k]] PISOT(16, 256 k + 83) = [[16, 256 k + 83], [5 + 16 k, 1 + 3 k]] 2 PISOT(16, 256 k + 84) = [[16, 256 k + 84, 4096 k + 2688 k + 441, 3 2 65536 k + 64512 k + 21168 k + 2315], [5 + 16 k, 1 + 4 k, 2 + 5 k, -2 - 6 k]] PISOT(16, 256 k + 85) = 2 [[16, 256 k + 85, 4096 k + 2720 k + 452], [6 + 16 k, -4 - 11 k, 2 + 6 k]] PISOT(16, 256 k + 86) = [[16, 256 k + 86], [5 + 16 k, 2 + 6 k]] 2 PISOT(16, 256 k + 87) = [[16, 256 k + 87, 4096 k + 2784 k + 473, 3 2 65536 k + 66816 k + 22705 k + 2572], [5 + 16 k, 3 + 7 k, -3 - 10 k, -2 - 6 k]] 2 PISOT(16, 256 k + 95) = [[16, 256 k + 95, 4096 k + 3040 k + 564, 3 2 65536 k + 72960 k + 27073 k + 3348, 4 3 2 1048576 k + 1556480 k + 866352 k + 214296 k + 19874], [6 + 16 k, -k, -2 - 6 k, -2 - 4 k, 3 + 8 k]] 2 PISOT(16, 256 k + 97) = [[16, 256 k + 97, 4096 k + 3104 k + 588, 3 2 65536 k + 74496 k + 28225 k + 3564, 4 3 2 1048576 k + 1589248 k + 903216 k + 228120 k + 21602], [6 + 16 k, k, 2 + 6 k, 1 + 4 k, 3 + 8 k]] PISOT(16, 256 k + 99) = [[16, 256 k + 99], [7 + 16 k, -5 - 13 k]] PISOT(16, 256 k + 101) = [[16, 256 k + 101], [6 + 16 k, 2 + 5 k]] PISOT(16, 256 k + 102) = [[16, 256 k + 102], [7 + 16 k, -4 - 10 k]] PISOT(16, 256 k + 110) = [[16, 256 k + 110], [6 + 16 k, 6 + 14 k]] 2 PISOT(16, 256 k + 111) = [[16, 256 k + 111, 4096 k + 3552 k + 770, 3 2 65536 k + 85248 k + 36961 k + 5341], [6 + 16 k, 6 + 15 k, 3 + 8 k, 3 + 7 k]] PISOT(16, 256 k + 113) = 2 [[16, 256 k + 113, 4096 k + 3616 k + 798], [7 + 16 k, 1 + k, -4 - 9 k]] PISOT(16, 256 k + 117) = [[16, 256 k + 117], [8 + 16 k, -5 - 11 k]] 2 PISOT(16, 256 k + 124) = [[16, 256 k + 124, 4096 k + 3968 k + 961, 3 2 65536 k + 95232 k + 46128 k + 7448, 4 3 2 1048576 k + 2031616 k + 1476096 k + 476664 k + 57724], [8 + 16 k, -2 - 4 k, 1 + k, -4 - 8 k, 1 + 2 k]] PISOT(16, 256 k + 126) = [[16, 256 k + 126], [8 + 16 k, -1 - 2 k]] PISOT(16, 256 k + 127) = 2 [[16, 256 k + 127, 4096 k + 4064 k + 1008], [8 + 16 k, -1 - k, 4 + 8 k]] PISOT(16, 256 k + 129) = 2 [[16, 256 k + 129, 4096 k + 4128 k + 1040], [8 + 16 k, k, 4 + 8 k]] PISOT(16, 256 k + 130) = [[16, 256 k + 130], [8 + 16 k, 1 + 2 k]] 2 PISOT(16, 256 k + 132) = [[16, 256 k + 132, 4096 k + 4224 k + 1089, 3 2 65536 k + 101376 k + 52272 k + 8984, 4 3 2 1048576 k + 2162688 k + 1672704 k + 574984 k + 74116], [8 + 16 k, 2 + 4 k, 1 + k, -4 - 8 k, -1 - 2 k]] PISOT(16, 256 k + 139) = [[16, 256 k + 139], [8 + 16 k, 6 + 11 k]] PISOT(16, 256 k + 143) = 2 [[16, 256 k + 143, 4096 k + 4576 k + 1278], [9 + 16 k, -k, -5 - 9 k]] 2 PISOT(16, 256 k + 145) = [[16, 256 k + 145, 4096 k + 4640 k + 1314, 3 2 65536 k + 111360 k + 63073 k + 11908], [10 + 16 k, -9 - 15 k, 5 + 8 k, -4 - 7 k]] PISOT(16, 256 k + 146) = [[16, 256 k + 146], [10 + 16 k, -8 - 14 k]] PISOT(16, 256 k + 154) = [[16, 256 k + 154], [9 + 16 k, 6 + 10 k]] PISOT(16, 256 k + 155) = [[16, 256 k + 155], [10 + 16 k, -3 - 5 k]] PISOT(16, 256 k + 157) = [[16, 256 k + 157], [9 + 16 k, 8 + 13 k]] 2 PISOT(16, 256 k + 159) = [[16, 256 k + 159, 4096 k + 5088 k + 1580, 3 2 65536 k + 122112 k + 75841 k + 15701, 4 3 2 1048576 k + 2605056 k + 2426928 k + 1004872 k + 156026], [10 + 16 k, -1 - k, 4 + 6 k, -3 - 4 k, 5 + 8 k]] 2 PISOT(16, 256 k + 161) = [[16, 256 k + 161, 4096 k + 5152 k + 1620, 3 2 65536 k + 123648 k + 77761 k + 16301, 4 3 2 1048576 k + 2637824 k + 2488368 k + 1043272 k + 164026], [10 + 16 k, 1 + k, -4 - 6 k, 2 + 4 k, 5 + 8 k]] 2 PISOT(16, 256 k + 169) = [[16, 256 k + 169, 4096 k + 5408 k + 1785, 3 2 65536 k + 129792 k + 85681 k + 18853], [11 + 16 k, -4 - 7 k, -7 - 10 k, 4 + 6 k]] PISOT(16, 256 k + 170) = [[16, 256 k + 170], [11 + 16 k, -4 - 6 k]] PISOT(16, 256 k + 171) = [ 2 [16, 256 k + 171, 4096 k + 5472 k + 1828], [10 + 16 k, 7 + 11 k, 4 + 6 k]] 2 PISOT(16, 256 k + 172) = [[16, 256 k + 172, 4096 k + 5504 k + 1849, 3 2 65536 k + 132096 k + 88752 k + 19877], [11 + 16 k, -3 - 4 k, 3 + 5 k, 4 + 6 k]] PISOT(16, 256 k + 173) = [[16, 256 k + 173], [11 + 16 k, -2 - 3 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998474997380933 2 PISOT(16, 256 k + 174) = [[16, 256 k + 174, 4096 k + 5568 k + 1892, 3 2 65536 k + 133632 k + 90820 k + 20573], [12 + 16 k, -13 - 18 k, 8 + 12 k, 2 + 3 k]] 2 PISOT(16, 256 k + 179) = [[16, 256 k + 179, 4096 k + 5728 k + 2003, 3 2 65536 k + 137472 k + 96137 k + 22413], [11 + 16 k, 2 + 3 k, 2 + 2 k, -7 - 10 k]] 2 PISOT(16, 256 k + 182) = [[16, 256 k + 182, 4096 k + 5824 k + 2070, 3 2 65536 k + 139776 k + 99364 k + 23543, 4 3 2 5 1048576 k + 2981888 k + 3179712 k + 1506856 k + 267765, 16777216 k 4 3 2 + 59637760 k + 84793344 k + 60276632 k + 21423032 k + 3045410], [11 + 16 k, 4 + 6 k, 3 + 4 k, -2 - 3 k, -1 - 2 k, -5 - 7 k]] PISOT(16, 256 k + 183) = [[16, 256 k + 183], [11 + 16 k, 5 + 7 k]] 2 PISOT(16, 256 k + 184) = [[16, 256 k + 184, 4096 k + 5888 k + 2116, 3 2 65536 k + 141312 k + 101568 k + 24334, 4 3 2 1048576 k + 3014656 k + 3250176 k + 1557376 k + 279841], [12 + 16 k, -6 - 8 k, 3 + 4 k, -1 - 2 k, -5 - 7 k]] 2 PISOT(16, 256 k + 186) = [[16, 256 k + 186, 4096 k + 5952 k + 2162, 3 2 65536 k + 142848 k + 103780 k + 25130, 4 3 2 5 1048576 k + 3047424 k + 3321024 k + 1608424 k + 292098, 16777216 k 4 3 2 + 60948480 k + 88561664 k + 64339368 k + 23369740 k + 3395195], [12 + 16 k, -4 - 6 k, -4 - 6 k, -4 - 6 k, -5 - 6 k, 8 + 11 k]] 2 PISOT(16, 256 k + 187) = [[16, 256 k + 187, 4096 k + 5984 k + 2186], [13 + 16 k, -16 - 21 k, 8 + 11 k]] 2 PISOT(16, 256 k + 190) = [[16, 256 k + 190, 4096 k + 6080 k + 2256, 3 2 65536 k + 145920 k + 108292 k + 26787, 4 3 2 1048576 k + 3112960 k + 3465408 k + 1714464 k + 318060], [12 + 16 k, -2 - 2 k, 6 + 8 k, -1 - k, 3 + 4 k]] PISOT(16, 256 k + 191) = 2 [[16, 256 k + 191, 4096 k + 6112 k + 2280], [12 + 16 k, -1 - k, 3 + 4 k]] PISOT(16, 256 k + 193) = 2 [[16, 256 k + 193, 4096 k + 6176 k + 2328], [12 + 16 k, 1 + k, -3 - 4 k]] 2 PISOT(16, 256 k + 194) = [[16, 256 k + 194, 4096 k + 6208 k + 2352, 3 2 65536 k + 148992 k + 112900 k + 28515, 4 3 2 1048576 k + 3178496 k + 3612864 k + 1825056 k + 345708], [12 + 16 k, 2 + 2 k, -6 - 8 k, -1 - k, 3 + 4 k]] 2 PISOT(16, 256 k + 198) = [[16, 256 k + 198, 4096 k + 6336 k + 2450, 3 2 65536 k + 152064 k + 117604 k + 30316, 4 3 2 5 1048576 k + 3244032 k + 3763392 k + 1940312 k + 375126, 16777216 k 4 3 2 + 64880640 k + 100358144 k + 77614680 k + 30011668 k + 4641757], [12 + 16 k, 4 + 6 k, 8 + 10 k, -3 - 4 k, -1 - 2 k, -7 - 9 k]] PISOT(16, 256 k + 199) = [[16, 256 k + 199], [13 + 16 k, -7 - 9 k]] PISOT(16, 256 k + 205) = [ 2 [16, 256 k + 205, 4096 k + 6560 k + 2627], [13 + 16 k, -3 - 3 k, 8 + 10 k] ] 2 PISOT(16, 256 k + 212) = [[16, 256 k + 212, 4096 k + 6784 k + 2809, 3 2 65536 k + 162816 k + 134832 k + 37219, 4 3 2 1048576 k + 3473408 k + 4314624 k + 2382024 k + 493148], [13 + 16 k, 3 + 4 k, 4 + 5 k, 2 + 2 k, -5 - 6 k]] PISOT(16, 256 k + 214) = [[16, 256 k + 214], [13 + 16 k, 5 + 6 k]] PISOT(16, 256 k + 219) = [ 2 [16, 256 k + 219, 4096 k + 7008 k + 2998], [13 + 16 k, 9 + 11 k, 6 + 7 k]] PISOT(16, 256 k + 220) = [ 2 [16, 256 k + 220, 4096 k + 7040 k + 3025], [14 + 16 k, -3 - 4 k, -6 - 7 k] ] 2 PISOT(16, 256 k + 221) = [[16, 256 k + 221, 4096 k + 7072 k + 3053, 3 2 65536 k + 169728 k + 146537 k + 42176, 4 3 2 5 1048576 k + 3620864 k + 4689072 k + 2699058 k + 582645, 16777216 k 4 3 2 + 72417280 k + 125040128 k + 107957477 k + 46607241 k + 8049014], [15 + 16 k, -17 - 19 k, 9 + 10 k, -5 - 6 k, -3 - 3 k, 6 + 7 k]] 2 PISOT(16, 256 k + 223) = [[16, 256 k + 223, 4096 k + 7136 k + 3108, 3 2 65536 k + 171264 k + 149185 k + 43317, 4 3 2 1048576 k + 3653632 k + 4773936 k + 2772312 k + 603720], [14 + 16 k, -1 - k, 2 + 2 k, -4 - 4 k, 7 + 8 k]] 2 PISOT(16, 256 k + 225) = [[16, 256 k + 225, 4096 k + 7200 k + 3164, 3 2 65536 k + 172800 k + 151873 k + 44493, 4 3 2 1048576 k + 3686400 k + 4859952 k + 2847576 k + 625672], [14 + 16 k, 1 + k, -2 - 2 k, 3 + 4 k, 7 + 8 k]] PISOT(16, 256 k + 228) = 2 [[16, 256 k + 228, 4096 k + 7296 k + 3249], [14 + 16 k, 3 + 4 k, 8 + 9 k]] PISOT(16, 256 k + 230) = [[16, 256 k + 230], [15 + 16 k, -9 - 10 k]] 2 PISOT(16, 256 k + 231) = [[16, 256 k + 231, 4096 k + 7392 k + 3335, 3 2 65536 k + 177408 k + 160081 k + 48148, 4 3 2 5 1048576 k + 3784704 k + 5122608 k + 3081506 k + 695121, 16777216 k 4 3 2 + 75694080 k + 136603136 k + 123261015 k + 55610473 k + 10035582], [14 + 16 k, 6 + 7 k, 5 + 5 k, -7 - 8 k, -3 - 4 k, -9 - 10 k]] 2 PISOT(16, 256 k + 235) = [[16, 256 k + 235, 4096 k + 7520 k + 3452], [16 + 16 k, -20 - 21 k, 11 + 12 k]] PISOT(16, 256 k + 238) = [[16, 256 k + 238], [14 + 16 k, 13 + 14 k]] PISOT(16, 256 k + 239) = [[16, 256 k + 239], [14 + 16 k, 14 + 15 k]] 2 PISOT(16, 256 k + 247) = [[16, 256 k + 247, 4096 k + 7904 k + 3813, 3 2 65536 k + 189696 k + 183025 k + 58862, 4 3 2 5 1048576 k + 4046848 k + 5856816 k + 3767206 k + 908664, 16777216 k 4 3 2 + 80936960 k + 156182016 k + 150689095 k + 72694045 k + 14027221], [16 + 16 k, -9 - 9 k, 5 + 5 k, -3 - 3 k, 2 + 2 k, -1 - k]] 2 PISOT(16, 256 k + 248) = [[16, 256 k + 248, 4096 k + 7936 k + 3844, 3 2 65536 k + 190464 k + 184512 k + 59582, 4 3 2 1048576 k + 4063232 k + 5904384 k + 3813248 k + 923521], [16 + 16 k, -8 - 8 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(16, 256 k + 249) = [[16, 256 k + 249, 4096 k + 7968 k + 3875, 3 2 65536 k + 191232 k + 186001 k + 60304], [16 + 16 k, -7 - 7 k, 3 + 3 k, -1 - k]] 2 PISOT(16, 256 k + 250) = [[16, 256 k + 250, 4096 k + 8000 k + 3906, 3 2 65536 k + 192000 k + 187492 k + 61027], [16 + 16 k, -6 - 6 k, 2 + 2 k, -1 - k]] 2 PISOT(16, 256 k + 251) = [[16, 256 k + 251, 4096 k + 8032 k + 3938, 3 2 65536 k + 192768 k + 189017 k + 61784], [16 + 16 k, -5 - 5 k, 2 + 2 k, -1 - k]] PISOT(16, 256 k + 252) = 2 [[16, 256 k + 252, 4096 k + 8064 k + 3969], [16 + 16 k, -4 - 4 k, 1 + k]] PISOT(16, 256 k + 253) = 2 [[16, 256 k + 253, 4096 k + 8096 k + 4001], [16 + 16 k, -3 - 3 k, 1 + k]] PISOT(16, 256 k + 254) = [[16, 256 k + 254], [16 + 16 k, -2 - 2 k]] PISOT(16, 256 k + 255) = [[16, 256 k + 255], [16 + 16 k, -1 - k]] PISOT(17, 289 k + 1) = [[17, 289 k + 1], [17 k, k]] PISOT(17, 289 k + 2) = [[17, 289 k + 2], [17 k, 2 k]] 2 PISOT(17, 289 k + 3) = [[17, 289 k + 3, 4913 k + 102 k + 1], [17 k, 3 k, k]] 2 PISOT(17, 289 k + 4) = [[17, 289 k + 4, 4913 k + 136 k + 1], [17 k, 4 k, k]] 2 PISOT(17, 289 k + 5) = [[17, 289 k + 5, 4913 k + 170 k + 1], [17 k, 5 k, k]] PISOT(17, 289 k + 6) = [ 2 3 2 [17, 289 k + 6, 4913 k + 204 k + 2, 83521 k + 5202 k + 104 k + 1], [17 k, 6 k, 2 k, k]] PISOT(17, 289 k + 7) = [ 2 3 2 [17, 289 k + 7, 4913 k + 238 k + 3, 83521 k + 6069 k + 151 k + 1], [17 k, 7 k, 3 k, k]] 2 PISOT(17, 289 k + 8) = [[17, 289 k + 8, 4913 k + 272 k + 4, 3 2 83521 k + 6936 k + 200 k + 2, 4 3 2 1419857 k + 157216 k + 6732 k + 132 k + 1], [17 k, 8 k, 4 k, 2 k, k]] 2 PISOT(17, 289 k + 9) = [[17, 289 k + 9, 4913 k + 306 k + 5, 3 2 83521 k + 7803 k + 251 k + 3, 4 3 2 1419857 k + 176868 k + 8466 k + 192 k + 2, 5 4 3 2 24137569 k + 3758445 k + 238714 k + 7920 k + 147 k + 1], [1 + 17 k, -8 k, -4 k, -2 k, -k, -k]] 2 PISOT(17, 289 k + 11) = [[17, 289 k + 11, 4913 k + 374 k + 7, 3 2 83521 k + 9537 k + 359 k + 4, 4 3 2 1419857 k + 216172 k + 12240 k + 290 k + 2, 5 4 3 2 24137569 k + 4593655 k + 347378 k + 12653 k + 205 k + 1], [17 k, 11 k, 7 k, 4 k, 2 k, k]] PISOT(17, 289 k + 18) = [[17, 289 k + 18], [2 + 17 k, -1 - 16 k]] PISOT(17, 289 k + 19) = [[17, 289 k + 19], [2 + 17 k, -1 - 15 k]] 2 PISOT(17, 289 k + 30) = [[17, 289 k + 30, 4913 k + 1020 k + 53, 3 2 83521 k + 26010 k + 2702 k + 94, 4 3 2 1419857 k + 589560 k + 91851 k + 6376 k + 167], [1 + 17 k, 1 + 13 k, 1 + 6 k, -6 k, -1 - 10 k]] PISOT(17, 289 k + 31) = 2 [[17, 289 k + 31, 4913 k + 1054 k + 57], [1 + 17 k, 1 + 14 k, 1 + 9 k]] PISOT(17, 289 k + 32) = [ 2 3 2 [17, 289 k + 32, 4913 k + 1088 k + 60, 83521 k + 27744 k + 3064 k + 112] , [2 + 17 k, -2 k, -1 - 4 k, 1 + 9 k]] PISOT(17, 289 k + 36) = [ 2 3 2 [17, 289 k + 36, 4913 k + 1224 k + 76, 83521 k + 31212 k + 3880 k + 160] , [2 + 17 k, 2 k, 4 k, 1 + 8 k]] 2 PISOT(17, 289 k + 40) = [[17, 289 k + 40, 4913 k + 1360 k + 94, 3 2 83521 k + 34680 k + 4796 k + 221, 4 3 2 1419857 k + 786080 k + 163098 k + 15034 k + 520], [2 + 17 k, 1 + 6 k, -1 - 3 k, 1 + 10 k, 1 + 7 k]] PISOT(17, 289 k + 41) = [[17, 289 k + 41], [2 + 17 k, 1 + 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.979540393002514 2 PISOT(17, 289 k + 44) = [[17, 289 k + 44, 4913 k + 1496 k + 114, 3 2 83521 k + 38148 k + 5812 k + 295, 4 3 2 1419857 k + 864688 k + 197574 k + 20062 k + 763, 5 4 3 2 24137569 k + 18374620 k + 5597352 k + 852581 k + 64898 k + 1973], [3 + 17 k, -1 - 7 k, -k, -3 k, -2 - 8 k, 2 + 13 k]] PISOT(17, 289 k + 45) = [[ 2 3 2 17, 289 k + 45, 4913 k + 1530 k + 119, 83521 k + 39015 k + 6071 k + 315] , [2 + 17 k, 2 + 11 k, -5 k, -2 - 13 k]] PISOT(17, 289 k + 47) = 2 [[17, 289 k + 47, 4913 k + 1598 k + 130], [3 + 17 k, -1 - 4 k, 1 + 6 k]] PISOT(17, 289 k + 48) = 2 [[17, 289 k + 48, 4913 k + 1632 k + 136], [2 + 17 k, 2 + 14 k, 1 + 6 k]] PISOT(17, 289 k + 49) = 2 [[17, 289 k + 49, 4913 k + 1666 k + 141], [3 + 17 k, -2 k, -1 - 6 k]] PISOT(17, 289 k + 53) = 2 [[17, 289 k + 53, 4913 k + 1802 k + 165], [3 + 17 k, 1 + 2 k, -2 - 11 k]] PISOT(17, 289 k + 57) = [[ 2 3 2 17, 289 k + 57, 4913 k + 1938 k + 191, 83521 k + 49419 k + 9743 k + 640] , [3 + 17 k, 1 + 6 k, 3 k, 2 + 10 k]] PISOT(17, 289 k + 58) = [[17, 289 k + 58], [4 + 17 k, -2 - 10 k]] PISOT(17, 289 k + 64) = [[ 2 3 2 17, 289 k + 64, 4913 k + 2176 k + 241, 83521 k + 55488 k + 12290 k + 908 ], [4 + 17 k, -1 - 4 k, 1 + 2 k, -2 - 9 k]] PISOT(17, 289 k + 65) = 2 [[17, 289 k + 65, 4913 k + 2210 k + 249], [5 + 17 k, -5 - 20 k, 2 + 9 k]] PISOT(17, 289 k + 67) = 2 [[17, 289 k + 67, 4913 k + 2278 k + 264], [4 + 17 k, -1 - k, 3 + 13 k]] PISOT(17, 289 k + 71) = 2 [[17, 289 k + 71, 4913 k + 2414 k + 297], [4 + 17 k, 1 + 3 k, -1 - 4 k]] PISOT(17, 289 k + 72) = [[17, 289 k + 72], [4 + 17 k, 1 + 4 k]] PISOT(17, 289 k + 73) = 2 [[17, 289 k + 73, 4913 k + 2482 k + 313], [4 + 17 k, 1 + 5 k, 1 + 4 k]] 2 PISOT(17, 289 k + 74) = [[17, 289 k + 74, 4913 k + 2516 k + 322, 3 2 83521 k + 64158 k + 16424 k + 1401], [5 + 17 k, -3 - 11 k, 1 + 3 k, -1 - 4 k]] 2 PISOT(17, 289 k + 75) = [[17, 289 k + 75, 4913 k + 2550 k + 331, 3 2 83521 k + 65025 k + 16879 k + 1461, 4 3 2 1419857 k + 1473900 k + 573852 k + 99324 k + 6449, 5 4 3 2 24137569 k + 31320375 k + 16258562 k + 4220712 k + 547977 k + 28467], [5 + 17 k, -2 - 10 k, -3 - 10 k, 2 + 7 k, -1 - 3 k, 1 + 4 k]] PISOT(17, 289 k + 77) = 2 [[17, 289 k + 77, 4913 k + 2618 k + 349], [5 + 17 k, -3 - 8 k, 4 + 15 k]] PISOT(17, 289 k + 79) = [[17, 289 k + 79], [4 + 17 k, 3 + 11 k]] PISOT(17, 289 k + 82) = [[17, 289 k + 82], [4 + 17 k, 4 + 14 k]] PISOT(17, 289 k + 83) = 2 [[17, 289 k + 83, 4913 k + 2822 k + 405], [5 + 17 k, -1 - 2 k, 2 + 7 k]] PISOT(17, 289 k + 87) = 2 [[17, 289 k + 87, 4913 k + 2958 k + 445], [5 + 17 k, 2 k, 3 + 10 k]] PISOT(17, 289 k + 89) = [[17, 289 k + 89], [6 + 17 k, -4 - 13 k]] 2 PISOT(17, 289 k + 95) = [[17, 289 k + 95, 4913 k + 3230 k + 531, 3 2 83521 k + 82365 k + 27079 k + 2968], [5 + 17 k, 3 + 10 k, 2 + 5 k, -2 - 6 k]] PISOT(17, 289 k + 96) = [[17, 289 k + 96], [6 + 17 k, -2 - 6 k]] PISOT(17, 289 k + 97) = [[17, 289 k + 97], [5 + 17 k, 4 + 12 k]] 2 PISOT(17, 289 k + 98) = [[17, 289 k + 98, 4913 k + 3332 k + 565, 3 2 83521 k + 84966 k + 28814 k + 3257], [5 + 17 k, 4 + 13 k, 2 + 7 k, 2 + 6 k]] PISOT(17, 289 k + 99) = [[17, 289 k + 99], [6 + 17 k, -1 - 3 k]] PISOT(17, 289 k + 100) = 2 [[17, 289 k + 100, 4913 k + 3400 k + 588], [5 + 17 k, 5 + 15 k, 1 + 3 k]] 2 PISOT(17, 289 k + 108) = [[17, 289 k + 108, 4913 k + 3672 k + 686, 3 2 83521 k + 93636 k + 34988 k + 4357], [6 + 17 k, 2 + 6 k, 1 + 4 k, 3 + 8 k]] 2 PISOT(17, 289 k + 109) = [[17, 289 k + 109, 4913 k + 3706 k + 699, 3 2 83521 k + 94503 k + 35647 k + 4483, 4 3 2 1419857 k + 2142068 k + 1211964 k + 304804 k + 28751], [6 + 17 k, 3 + 7 k, -2 - 6 k, -2 - 4 k, 3 + 8 k]] PISOT(17, 289 k + 114) = [[17, 289 k + 114], [7 + 17 k, -2 - 5 k]] PISOT(17, 289 k + 115) = [ 2 [17, 289 k + 115, 4913 k + 3910 k + 778], [7 + 17 k, -1 - 4 k, -4 - 10 k]] 2 PISOT(17, 289 k + 117) = [[17, 289 k + 117, 4913 k + 3978 k + 805, 3 2 83521 k + 101439 k + 41059 k + 5539], [6 + 17 k, 5 + 15 k, 7 + 18 k, 2 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.986496331876077 2 PISOT(17, 289 k + 123) = [[17, 289 k + 123, 4913 k + 4182 k + 890, 3 2 83521 k + 106641 k + 45389 k + 6440], [8 + 17 k, -6 - 13 k, 3 + 8 k, 3 + 7 k]] PISOT(17, 289 k + 124) = [ 2 [17, 289 k + 124, 4913 k + 4216 k + 904], [8 + 17 k, -6 - 12 k, 6 + 14 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.976253041037944 2 PISOT(17, 289 k + 126) = [[17, 289 k + 126, 4913 k + 4284 k + 934, 3 2 83521 k + 109242 k + 47632 k + 6923, 4 3 2 1419857 k + 2476152 k + 1619454 k + 470750 k + 51315, 5 4 3 2 24137569 k + 52618230 k + 45883952 k + 20006385 k + 4361662 k + 380360] , [8 + 17 k, -4 - 10 k, -3 - 6 k, 3 + 6 k, -3 - 6 k, 3 + 7 k]] PISOT(17, 289 k + 128) = [[17, 289 k + 128], [7 + 17 k, 4 + 9 k]] 2 PISOT(17, 289 k + 132) = [[17, 289 k + 132, 4913 k + 4488 k + 1025, 3 2 83521 k + 114444 k + 52274 k + 7959], [8 + 17 k, -2 - 4 k, 2 + 3 k, -5 - 11 k]] 2 PISOT(17, 289 k + 139) = [[17, 289 k + 139, 4913 k + 4726 k + 1137, 3 2 83521 k + 120513 k + 57979 k + 9300, 4 3 2 1419857 k + 2731628 k + 1971150 k + 632286 k + 76069], [8 + 17 k, 2 + 3 k, -4 - 9 k, -3 - 6 k, 1 + 2 k]] PISOT(17, 289 k + 141) = [ 2 [17, 289 k + 141, 4913 k + 4794 k + 1169], [9 + 17 k, -6 - 12 k, 1 + 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998381804829322 2 PISOT(17, 289 k + 142) = [[17, 289 k + 142, 4913 k + 4828 k + 1186, 3 2 83521 k + 123114 k + 60488 k + 9906], [8 + 17 k, 2 + 6 k, 8 + 16 k, -1 - 2 k]] 2 PISOT(17, 289 k + 143) = [[17, 289 k + 143, 4913 k + 4862 k + 1203, 3 2 83521 k + 123981 k + 61351 k + 10120], [9 + 17 k, -5 - 10 k, k, 4 + 8 k]] PISOT(17, 289 k + 144) = [[17, 289 k + 144], [8 + 17 k, 4 + 8 k]] PISOT(17, 289 k + 145) = [[17, 289 k + 145], [9 + 17 k, -4 - 8 k]] 2 PISOT(17, 289 k + 146) = [[17, 289 k + 146, 4913 k + 4964 k + 1254, 3 2 83521 k + 126582 k + 63952 k + 10771], [8 + 17 k, 5 + 10 k, 1 + k, -4 - 8 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998380144090446 2 PISOT(17, 289 k + 147) = [[17, 289 k + 147, 4913 k + 4998 k + 1271, 3 2 83521 k + 127449 k + 64823 k + 10989], [9 + 17 k, -4 - 6 k, 8 + 16 k, 1 + 2 k]] PISOT(17, 289 k + 148) = 2 [[17, 289 k + 148, 4913 k + 5032 k + 1288], [8 + 17 k, 6 + 12 k, 1 + 2 k]] 2 PISOT(17, 289 k + 150) = [[17, 289 k + 150, 4913 k + 5100 k + 1324, 3 2 83521 k + 130050 k + 67516 k + 11687, 4 3 2 1419857 k + 2947800 k + 2295408 k + 794558 k + 103162], [9 + 17 k, -1 - 3 k, -5 - 9 k, 3 + 6 k, 1 + 2 k]] 2 PISOT(17, 289 k + 157) = [[17, 289 k + 157, 4913 k + 5338 k + 1450, 3 2 83521 k + 136119 k + 73949 k + 13392], [9 + 17 k, 2 + 4 k, 1 + 3 k, 6 + 11 k]] PISOT(17, 289 k + 161) = [[17, 289 k + 161], [10 + 17 k, -5 - 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.976253392445487 2 PISOT(17, 289 k + 163) = [[17, 289 k + 163, 4913 k + 5542 k + 1563, 3 2 83521 k + 141321 k + 79711 k + 14988, 4 3 2 5 1419857 k + 3203276 k + 2710140 k + 1019130 k + 143724, 24137569 k 4 3 2 + 68069615 k + 76786722 k + 43311781 k + 12215673 k + 1378208], [9 + 17 k, 6 + 10 k, -3 - 6 k, -3 - 6 k, -3 - 6 k, -4 - 7 k]] PISOT(17, 289 k + 165) = [ 2 [17, 289 k + 165, 4913 k + 5610 k + 1601], [9 + 17 k, 6 + 12 k, 8 + 14 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.986494936069970 2 PISOT(17, 289 k + 166) = [[17, 289 k + 166, 4913 k + 5644 k + 1621, 3 2 83521 k + 143922 k + 82670 k + 15829], [9 + 17 k, 7 + 13 k, 5 + 8 k, -4 - 7 k]] 2 PISOT(17, 289 k + 172) = [[17, 289 k + 172, 4913 k + 5848 k + 1740, 3 2 83521 k + 149124 k + 88744 k + 17602], [11 + 17 k, -10 - 15 k, 11 + 18 k, -3 - 5 k]] PISOT(17, 289 k + 174) = [ 2 [17, 289 k + 174, 4913 k + 5916 k + 1781], [10 + 17 k, 3 + 4 k, -6 - 10 k] ] PISOT(17, 289 k + 175) = [[17, 289 k + 175], [10 + 17 k, 3 + 5 k]] 2 PISOT(17, 289 k + 180) = [[17, 289 k + 180, 4913 k + 6120 k + 1906, 3 2 83521 k + 156060 k + 97204 k + 20182, 4 3 2 1419857 k + 3537360 k + 3304902 k + 1372348 k + 213700], [11 + 17 k, -4 - 7 k, -4 - 6 k, 2 + 4 k, 5 + 8 k]] 2 PISOT(17, 289 k + 181) = [[17, 289 k + 181, 4913 k + 6154 k + 1927, 3 2 83521 k + 156927 k + 98279 k + 20516], [11 + 17 k, -4 - 6 k, 3 + 4 k, -5 - 8 k]] 2 PISOT(17, 289 k + 189) = [[17, 289 k + 189, 4913 k + 6426 k + 2101], [12 + 17 k, -10 - 15 k, 2 + 3 k]] PISOT(17, 289 k + 190) = [[17, 289 k + 190], [11 + 17 k, 2 + 3 k]] 2 PISOT(17, 289 k + 191) = [[17, 289 k + 191, 4913 k + 6494 k + 2146, 3 2 83521 k + 165597 k + 109445 k + 24112], [12 + 17 k, -9 - 13 k, 5 + 7 k, -4 - 6 k]] PISOT(17, 289 k + 192) = [[17, 289 k + 192], [12 + 17 k, -8 - 12 k]] PISOT(17, 289 k + 193) = [[17, 289 k + 193], [11 + 17 k, 4 + 6 k]] 2 PISOT(17, 289 k + 194) = [[17, 289 k + 194, 4913 k + 6596 k + 2214, 3 2 83521 k + 168198 k + 112912 k + 25267], [12 + 17 k, -7 - 10 k, 3 + 5 k, 4 + 6 k]] PISOT(17, 289 k + 200) = [[17, 289 k + 200], [11 + 17 k, 9 + 13 k]] PISOT(17, 289 k + 202) = [ 2 [17, 289 k + 202, 4913 k + 6868 k + 2400], [12 + 17 k, -2 - 2 k, 7 + 10 k] ] PISOT(17, 289 k + 206) = 2 [[17, 289 k + 206, 4913 k + 7004 k + 2496], [12 + 17 k, 1 + 2 k, 5 + 7 k]] PISOT(17, 289 k + 207) = [[17, 289 k + 207], [13 + 17 k, -10 - 14 k]] PISOT(17, 289 k + 210) = [[17, 289 k + 210], [13 + 17 k, -8 - 11 k]] PISOT(17, 289 k + 212) = [ 2 [17, 289 k + 212, 4913 k + 7208 k + 2644], [12 + 17 k, 5 + 8 k, 11 + 15 k] ] 2 PISOT(17, 289 k + 214) = [[17, 289 k + 214, 4913 k + 7276 k + 2694, 3 2 83521 k + 185538 k + 137392 k + 33914, 4 3 2 5 1419857 k + 4205528 k + 4671294 k + 2306108 k + 426934, 24137569 k 4 3 2 + 89367470 k + 132352752 k + 98008414 k + 36288584 k + 5374554], [12 + 17 k, 8 + 10 k, -7 - 10 k, -5 - 7 k, -2 - 3 k, -3 - 4 k]] 2 PISOT(17, 289 k + 215) = [[17, 289 k + 215, 4913 k + 7310 k + 2719, 3 2 83521 k + 186405 k + 138671 k + 34386], [12 + 17 k, 8 + 11 k, 2 + 3 k, 3 + 4 k]] PISOT(17, 289 k + 216) = [ 2 [17, 289 k + 216, 4913 k + 7344 k + 2744], [13 + 17 k, -4 - 5 k, 3 + 4 k]] PISOT(17, 289 k + 217) = [[17, 289 k + 217], [13 + 17 k, -3 - 4 k]] PISOT(17, 289 k + 218) = [ 2 [17, 289 k + 218, 4913 k + 7412 k + 2796], [13 + 17 k, -2 - 3 k, -3 - 4 k] ] PISOT(17, 289 k + 222) = 2 [[17, 289 k + 222, 4913 k + 7548 k + 2899], [13 + 17 k, k, 10 + 13 k]] PISOT(17, 289 k + 224) = [ 2 [17, 289 k + 224, 4913 k + 7616 k + 2952], [12 + 17 k, 15 + 20 k, 7 + 9 k] ] 2 PISOT(17, 289 k + 225) = [[17, 289 k + 225, 4913 k + 7650 k + 2978, 3 2 83521 k + 195075 k + 151877 k + 39415], [13 + 17 k, 3 + 4 k, 1 + 2 k, 7 + 9 k]] PISOT(17, 289 k + 231) = [[17, 289 k + 231], [13 + 17 k, 8 + 10 k]] 2 PISOT(17, 289 k + 232) = [[17, 289 k + 232, 4913 k + 7888 k + 3166, 3 2 83521 k + 201144 k + 161468 k + 43205], [14 + 17 k, -5 - 6 k, 3 + 3 k, -8 - 10 k]] 2 PISOT(17, 289 k + 236) = [[17, 289 k + 236, 4913 k + 8024 k + 3276], [14 + 17 k, -1 - 2 k, -9 - 11 k]] PISOT(17, 289 k + 240) = [ 2 [17, 289 k + 240, 4913 k + 8160 k + 3388], [14 + 17 k, 2 + 2 k, -5 - 6 k]] 2 PISOT(17, 289 k + 241) = [[17, 289 k + 241, 4913 k + 8194 k + 3417], [15 + 17 k, -12 - 14 k, 5 + 6 k]] PISOT(17, 289 k + 242) = 2 [[17, 289 k + 242, 4913 k + 8228 k + 3445], [14 + 17 k, 3 + 4 k, 5 + 6 k]] 2 PISOT(17, 289 k + 244) = [[17, 289 k + 244, 4913 k + 8296 k + 3502, 3 2 83521 k + 211548 k + 178604 k + 50262], [15 + 17 k, -9 - 11 k, -5 - 5 k, 11 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.979539819317329 2 PISOT(17, 289 k + 245) = [[17, 289 k + 245, 4913 k + 8330 k + 3531, 3 2 83521 k + 212415 k + 180079 k + 50890, 4 3 2 5 1419857 k + 4814740 k + 6122652 k + 3460450 k + 733444, 24137569 k 4 3 2 + 102313225 k + 173474562 k + 147067445 k + 62341157 k + 10570645], [14 + 17 k, 6 + 7 k, -1 - k, 3 + 3 k, -6 - 8 k, -11 - 13 k]] PISOT(17, 289 k + 248) = [[17, 289 k + 248], [15 + 17 k, -6 - 7 k]] 2 PISOT(17, 289 k + 249) = [[17, 289 k + 249, 4913 k + 8466 k + 3647, 3 2 83521 k + 215883 k + 185999 k + 53416, 4 3 2 1419857 k + 4893348 k + 6324000 k + 3632350 k + 782361], [15 + 17 k, -5 - 6 k, -2 - 3 k, -9 - 10 k, 6 + 7 k]] 2 PISOT(17, 289 k + 253) = [[17, 289 k + 253, 4913 k + 8602 k + 3765, 3 2 83521 k + 219351 k + 192019 k + 56029], [15 + 17 k, -2 - 2 k, 4 + 4 k, -7 - 8 k]] 2 PISOT(17, 289 k + 257) = [[17, 289 k + 257, 4913 k + 8738 k + 3885, 3 2 83521 k + 222819 k + 198139 k + 58729], [15 + 17 k, 2 + 2 k, -3 - 4 k, -8 - 9 k]] 2 PISOT(17, 289 k + 258) = [[17, 289 k + 258, 4913 k + 8772 k + 3916], [16 + 17 k, -13 - 14 k, 8 + 9 k]] 2 PISOT(17, 289 k + 259) = [[17, 289 k + 259, 4913 k + 8806 k + 3946, 3 2 83521 k + 224553 k + 201245 k + 60119, 4 3 2 1419857 k + 5089868 k + 6842313 k + 4088074 k + 915939], [16 + 17 k, -12 - 13 k, 5 + 6 k, 6 + 6 k, -9 - 10 k]] PISOT(17, 289 k + 270) = [[17, 289 k + 270], [15 + 17 k, 14 + 15 k]] PISOT(17, 289 k + 271) = [[17, 289 k + 271], [15 + 17 k, 15 + 16 k]] 2 PISOT(17, 289 k + 278) = [[17, 289 k + 278, 4913 k + 9452 k + 4546, 3 2 83521 k + 241026 k + 231848 k + 74339, 4 3 2 5 1419857 k + 5463256 k + 7882866 k + 5055102 k + 1215637, 24137569 k 4 3 2 + 116094190 k + 223348448 k + 214843241 k + 103330258 k + 19878843], [17 + 17 k, -11 - 11 k, 7 + 7 k, -4 - 4 k, 2 + 2 k, -1 - k]] 2 PISOT(17, 289 k + 280) = [[17, 289 k + 280, 4913 k + 9520 k + 4612, 3 2 83521 k + 242760 k + 235208 k + 75966, 4 3 2 5 1419857 k + 5502560 k + 7997004 k + 5165564 k + 1251265, 24137569 k 4 3 2 + 116929400 k + 226580624 k + 219533242 k + 106354514 k + 20610064], [16 + 17 k, 8 + 8 k, -4 - 4 k, 2 + 2 k, -1 - k, 1 + k]] 2 PISOT(17, 289 k + 281) = [[17, 289 k + 281, 4913 k + 9554 k + 4645, 3 2 83521 k + 243627 k + 236891 k + 76783, 4 3 2 1419857 k + 5522212 k + 8054226 k + 5221112 k + 1269242], [17 + 17 k, -8 - 8 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(17, 289 k + 282) = [[17, 289 k + 282, 4913 k + 9588 k + 4678, 3 2 83521 k + 244494 k + 238576 k + 77602], [17 + 17 k, -7 - 7 k, 3 + 3 k, -1 - k]] 2 PISOT(17, 289 k + 283) = [[17, 289 k + 283, 4913 k + 9622 k + 4711, 3 2 83521 k + 245361 k + 240263 k + 78422], [17 + 17 k, -6 - 6 k, 2 + 2 k, -1 - k]] PISOT(17, 289 k + 284) = 2 [[17, 289 k + 284, 4913 k + 9656 k + 4744], [17 + 17 k, -5 - 5 k, 1 + k]] PISOT(17, 289 k + 285) = 2 [[17, 289 k + 285, 4913 k + 9690 k + 4778], [17 + 17 k, -4 - 4 k, 1 + k]] PISOT(17, 289 k + 286) = 2 [[17, 289 k + 286, 4913 k + 9724 k + 4812], [17 + 17 k, -3 - 3 k, 1 + k]] PISOT(17, 289 k + 287) = [[17, 289 k + 287], [17 + 17 k, -2 - 2 k]] PISOT(17, 289 k + 288) = [[17, 289 k + 288], [17 + 17 k, -1 - k]] PISOT(18, 324 k + 1) = [[18, 324 k + 1], [18 k, k]] PISOT(18, 324 k + 2) = [[18, 324 k + 2], [18 k, 2 k]] 2 PISOT(18, 324 k + 3) = [[18, 324 k + 3, 5832 k + 108 k + 1], [18 k, 3 k, k]] 2 PISOT(18, 324 k + 4) = [[18, 324 k + 4, 5832 k + 144 k + 1], [18 k, 4 k, k]] 2 PISOT(18, 324 k + 5) = [[18, 324 k + 5, 5832 k + 180 k + 1], [18 k, 5 k, k]] PISOT(18, 324 k + 6) = [ 2 3 2 [18, 324 k + 6, 5832 k + 216 k + 2, 104976 k + 5832 k + 108 k + 1], [18 k, 6 k, 2 k, k]] PISOT(18, 324 k + 7) = [ 2 3 2 [18, 324 k + 7, 5832 k + 252 k + 3, 104976 k + 6804 k + 157 k + 1], [18 k, 7 k, 3 k, k]] 2 PISOT(18, 324 k + 8) = [[18, 324 k + 8, 5832 k + 288 k + 4, 3 2 104976 k + 7776 k + 208 k + 2, 4 3 2 1889568 k + 186624 k + 7344 k + 136 k + 1], [18 k, 8 k, 4 k, 2 k, k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970419968581249 2 PISOT(18, 324 k + 9) = [[18, 324 k + 9, 5832 k + 324 k + 5, 3 2 104976 k + 8748 k + 261 k + 3, 4 3 2 1889568 k + 209952 k + 9234 k + 198 k + 2, 5 4 3 2 34012224 k + 4723920 k + 274104 k + 8505 k + 151 k + 1], [1 + 18 k, -9 k, -4 k, -2 k, -k, -k]] 2 PISOT(18, 324 k + 11) = [[18, 324 k + 11, 5832 k + 396 k + 7, 3 2 104976 k + 10692 k + 373 k + 4, 4 3 2 1889568 k + 256608 k + 13338 k + 298 k + 2, 5 4 3 2 34012224 k + 5773680 k + 398520 k + 13535 k + 209 k + 1], [18 k, 11 k, 7 k, 4 k, 2 k, k]] PISOT(18, 324 k + 19) = [[18, 324 k + 19], [2 + 18 k, -1 - 17 k]] PISOT(18, 324 k + 20) = [[18, 324 k + 20], [2 + 18 k, -1 - 16 k]] PISOT(18, 324 k + 29) = [[18, 324 k + 29], [1 + 18 k, 1 + 11 k]] PISOT(18, 324 k + 39) = [[ 2 3 2 18, 324 k + 39, 5832 k + 1404 k + 85, 104976 k + 37908 k + 4581 k + 185] , [3 + 18 k, -2 - 15 k, 4 k, 1 + 8 k]] PISOT(18, 324 k + 41) = 2 [[18, 324 k + 41, 5832 k + 1476 k + 93], [1 + 18 k, 2 + 23 k, 2 + 16 k]] PISOT(18, 324 k + 46) = [[ 2 3 2 18, 324 k + 46, 5832 k + 1656 k + 118, 104976 k + 44712 k + 6364 k + 303 ], [2 + 18 k, 2 + 10 k, -1 - 10 k, -1 - 7 k]] PISOT(18, 324 k + 47) = [[18, 324 k + 47], [3 + 18 k, -1 - 7 k]] PISOT(18, 324 k + 55) = [[ 2 3 2 18, 324 k + 55, 5832 k + 1980 k + 168, 104976 k + 53460 k + 9073 k + 513 ], [4 + 18 k, -3 - 17 k, 2 k, 1 + 6 k]] PISOT(18, 324 k + 59) = 2 [[18, 324 k + 59, 5832 k + 2124 k + 193], [4 + 18 k, -3 - 13 k, 2 + 11 k]] PISOT(18, 324 k + 64) = [[18, 324 k + 64], [3 + 18 k, 2 + 10 k]] 2 PISOT(18, 324 k + 65) = [[18, 324 k + 65, 5832 k + 2340 k + 235, 3 2 104976 k + 63180 k + 12685 k + 850, 4 3 2 1889568 k + 1516320 k + 456570 k + 61150 k + 3074], [4 + 18 k, -1 - 7 k, -1 - 7 k, -2 - 7 k, 2 + 10 k]] PISOT(18, 324 k + 75) = 2 [[18, 324 k + 75, 5832 k + 2700 k + 313], [4 + 18 k, 3 k, 3 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.9718266826 PISOT(18, 324 k + 76) = 2 [[18, 324 k + 76, 5832 k + 2736 k + 321], [4 + 18 k, 4 k, 4 + 17 k]] 2 PISOT(18, 324 k + 79) = [[18, 324 k + 79, 5832 k + 2844 k + 347, 3 2 104976 k + 76788 k + 18733 k + 1524], [4 + 18 k, 2 + 7 k, -1 - 5 k, -1 - 4 k]] PISOT(18, 324 k + 80) = [[18, 324 k + 80], [4 + 18 k, 2 + 8 k]] PISOT(18, 324 k + 81) = 2 [[18, 324 k + 81, 5832 k + 2916 k + 365], [5 + 18 k, -2 - 9 k, -1 - 4 k]] PISOT(18, 324 k + 82) = [[18, 324 k + 82], [5 + 18 k, -2 - 8 k]] PISOT(18, 324 k + 83) = 2 [[18, 324 k + 83, 5832 k + 2988 k + 383], [5 + 18 k, -2 - 7 k, 1 + 4 k]] 2 PISOT(18, 324 k + 84) = [[18, 324 k + 84, 5832 k + 3024 k + 392, 3 2 104976 k + 81648 k + 21168 k + 1829, 4 3 2 1889568 k + 1959552 k + 762048 k + 131700 k + 8534], [6 + 18 k, -7 - 24 k, 4 + 14 k, -2 - 7 k, 1 + 4 k]] 2 PISOT(18, 324 k + 89) = [[18, 324 k + 89, 5832 k + 3204 k + 440, 3 2 104976 k + 86508 k + 23761 k + 2175], [5 + 18 k, -k, -2 - 5 k, 3 + 11 k]] 2 PISOT(18, 324 k + 92) = [[18, 324 k + 92, 5832 k + 3312 k + 470, 3 2 104976 k + 89424 k + 25384 k + 2401], [4 + 18 k, 5 + 20 k, 3 + 12 k, 2 + 7 k]] PISOT(18, 324 k + 98) = [[18, 324 k + 98], [6 + 18 k, -3 - 10 k]] 2 PISOT(18, 324 k + 103) = [[18, 324 k + 103, 5832 k + 3708 k + 589, 3 2 104976 k + 100116 k + 31813 k + 3368, 4 3 2 1889568 k + 2402784 k + 1145394 k + 242582 k + 19259, 5 4 3 2 34012224 k + 54062640 k + 34364088 k + 10918459 k + 1734053 k + 110127] , [5 + 18 k, 4 + 13 k, 2 k, 3 + 11 k, 3 + 9 k, -1 - 3 k]] 2 PISOT(18, 324 k + 104) = [[18, 324 k + 104, 5832 k + 3744 k + 601, 3 2 104976 k + 101088 k + 32452 k + 3473, 4 3 2 1889568 k + 2426112 k + 1168236 k + 250036 k + 20069], [5 + 18 k, 5 + 14 k, -2 - 9 k, -5 - 16 k, -1 - 3 k]] 2 PISOT(18, 324 k + 106) = [[18, 324 k + 106, 5832 k + 3816 k + 624, 3 2 104976 k + 103032 k + 33700 k + 3673, 4 3 2 1889568 k + 2472768 k + 1213272 k + 264516 k + 21620], [6 + 18 k, -2 k, -4 - 12 k, k, 2 + 6 k]] PISOT(18, 324 k + 107) = 2 [[18, 324 k + 107, 5832 k + 3852 k + 636], [6 + 18 k, -k, -2 - 6 k]] PISOT(18, 324 k + 109) = 2 [[18, 324 k + 109, 5832 k + 3924 k + 660], [6 + 18 k, k, 2 + 6 k]] 2 PISOT(18, 324 k + 110) = [[18, 324 k + 110, 5832 k + 3960 k + 672, 3 2 104976 k + 106920 k + 36292 k + 4105, 4 3 2 1889568 k + 2566080 k + 1306584 k + 295620 k + 25076], [6 + 18 k, 2 k, 4 + 12 k, k, 2 + 6 k]] 2 PISOT(18, 324 k + 112) = [[18, 324 k + 112, 5832 k + 4032 k + 697, 3 2 104976 k + 108864 k + 37636 k + 4338], [5 + 18 k, 7 + 22 k, 4 + 11 k, -1 - 3 k]] 2 PISOT(18, 324 k + 116) = [[18, 324 k + 116, 5832 k + 4176 k + 748, 3 2 104976 k + 112752 k + 40384 k + 4823, 4 3 2 1889568 k + 2706048 k + 1453680 k + 347164 k + 31098], [6 + 18 k, 3 + 8 k, -1 - 2 k, 1 + 5 k, 5 + 14 k]] 2 PISOT(18, 324 k + 128) = [[18, 324 k + 128, 5832 k + 4608 k + 910, 3 2 104976 k + 124416 k + 49144 k + 6470, 4 3 2 1889568 k + 2985984 k + 1769256 k + 465880 k + 46001], [8 + 18 k, -6 - 16 k, -3 - 6 k, 5 + 12 k, -2 - 5 k]] PISOT(18, 324 k + 130) = 2 [[18, 324 k + 130, 5832 k + 4680 k + 939], [6 + 18 k, 8 + 22 k, 6 + 15 k]] PISOT(18, 324 k + 131) = [[18, 324 k + 131], [7 + 18 k, 2 + 5 k]] 2 PISOT(18, 324 k + 135) = [[18, 324 k + 135, 5832 k + 4860 k + 1013, 3 2 104976 k + 131220 k + 54693 k + 7601], [7 + 18 k, 4 + 9 k, -1 - 4 k, -5 - 12 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.980531854461797 2 PISOT(18, 324 k + 137) = [[18, 324 k + 137, 5832 k + 4932 k + 1043, 3 2 104976 k + 133164 k + 56317 k + 7941, 4 3 2 1889568 k + 3195936 k + 2027322 k + 571658 k + 60460, 5 4 3 2 34012224 k + 71908560 k + 60818040 k + 25722233 k + 5440243 k + 460321] , [8 + 18 k, -4 - 7 k, 8 + 19 k, k, 3 + 8 k, 3 + 7 k]] 2 PISOT(18, 324 k + 138) = [[18, 324 k + 138, 5832 k + 4968 k + 1058, 3 2 104976 k + 134136 k + 57132 k + 8111], [8 + 18 k, -3 - 6 k, 3 + 8 k, 3 + 7 k]] 2 PISOT(18, 324 k + 140) = [[18, 324 k + 140, 5832 k + 5040 k + 1089, 3 2 104976 k + 136080 k + 58804 k + 8471], [8 + 18 k, -1 - 4 k, -6 - 13 k, 3 + 7 k]] PISOT(18, 324 k + 142) = [[18, 324 k + 142], [7 + 18 k, 7 + 16 k]] 2 PISOT(18, 324 k + 143) = [[18, 324 k + 143, 5832 k + 5148 k + 1136, 3 2 104976 k + 138996 k + 61345 k + 9024, 4 3 2 1889568 k + 3335904 k + 2208438 k + 649760 k + 71684], [7 + 18 k, 7 + 17 k, 3 + 9 k, 7 + 17 k, 4 + 9 k]] PISOT(18, 324 k + 145) = 2 [[18, 324 k + 145, 5832 k + 5220 k + 1168], [7 + 18 k, 8 + 19 k, 4 + 9 k]] PISOT(18, 324 k + 147) = [ 2 [18, 324 k + 147, 5832 k + 5292 k + 1201], [8 + 18 k, 2 + 3 k, -5 - 11 k]] PISOT(18, 324 k + 148) = [ 2 [18, 324 k + 148, 5832 k + 5328 k + 1217], [9 + 18 k, -7 - 14 k, 5 + 11 k] ] PISOT(18, 324 k + 149) = [[18, 324 k + 149], [9 + 18 k, -6 - 13 k]] 2 PISOT(18, 324 k + 153) = [[18, 324 k + 153, 5832 k + 5508 k + 1301, 3 2 104976 k + 148716 k + 70245 k + 11063, 4 3 2 1889568 k + 3569184 k + 2528658 k + 796374 k + 94074, 5 4 3 2 34012224 k + 80306640 k + 75856824 k + 35832537 k + 8464543 k + 799956] , [9 + 18 k, -5 - 9 k, 7 + 14 k, -4 - 7 k, 6 + 13 k, 1 + 2 k]] 2 PISOT(18, 324 k + 156) = [[18, 324 k + 156, 5832 k + 5616 k + 1352, 3 2 104976 k + 151632 k + 73008 k + 11717, 4 3 2 1889568 k + 3639168 k + 2628288 k + 843636 k + 101544, 5 4 3 2 34012224 k + 81881280 k + 78848640 k + 37963836 k + 9139192 k + 880019] , [8 + 18 k, 5 + 12 k, 6 + 14 k, 6 + 13 k, 2 + 4 k, -1 - 2 k]] 2 PISOT(18, 324 k + 157) = [[18, 324 k + 157, 5832 k + 5652 k + 1369, 3 2 104976 k + 152604 k + 73933 k + 11937], [9 + 18 k, -2 - 5 k, -4 - 8 k, 1 + 2 k]] 2 PISOT(18, 324 k + 158) = [[18, 324 k + 158, 5832 k + 5688 k + 1387, 3 2 104976 k + 153576 k + 74896 k + 12176, 4 3 2 1889568 k + 3685824 k + 2696220 k + 876628 k + 106889], [8 + 18 k, 7 + 14 k, -1 - 3 k, -4 - 8 k, 1 + 2 k]] PISOT(18, 324 k + 160) = [[18, 324 k + 160], [9 + 18 k, -1 - 2 k]] PISOT(18, 324 k + 161) = 2 [[18, 324 k + 161, 5832 k + 5796 k + 1440], [8 + 18 k, 8 + 17 k, 4 + 8 k]] PISOT(18, 324 k + 163) = [ 2 [18, 324 k + 163, 5832 k + 5868 k + 1476], [10 + 18 k, -9 - 17 k, 4 + 8 k] ] PISOT(18, 324 k + 164) = [[18, 324 k + 164], [9 + 18 k, 1 + 2 k]] 2 PISOT(18, 324 k + 166) = [[18, 324 k + 166, 5832 k + 5976 k + 1531, 3 2 104976 k + 161352 k + 82672 k + 14120, 4 3 2 1889568 k + 3872448 k + 2976156 k + 1016612 k + 130225], [10 + 18 k, -7 - 14 k, -2 - 3 k, 4 + 8 k, 1 + 2 k]] 2 PISOT(18, 324 k + 167) = [[18, 324 k + 167, 5832 k + 6012 k + 1549, 3 2 104976 k + 162324 k + 83653 k + 14368], [9 + 18 k, 3 + 5 k, -4 - 8 k, -1 - 2 k]] 2 PISOT(18, 324 k + 168) = [[18, 324 k + 168, 5832 k + 6048 k + 1568, 3 2 104976 k + 163296 k + 84672 k + 14635, 4 3 2 5 1889568 k + 3919104 k + 3048192 k + 1053708 k + 136596, 34012224 k 4 3 2 + 88179840 k + 91445760 k + 47416644 k + 12293440 k + 1274921], [10 + 18 k, -7 - 12 k, 8 + 14 k, -7 - 13 k, 2 + 4 k, 1 + 2 k]] 2 PISOT(18, 324 k + 171) = [[18, 324 k + 171, 5832 k + 6156 k + 1625, 3 2 104976 k + 166212 k + 87741 k + 15442, 4 3 2 5 1889568 k + 3989088 k + 3158514 k + 1111662 k + 146742, 34012224 k 4 3 2 + 89754480 k + 94752504 k + 50020335 k + 13204501 k + 1394458], [9 + 18 k, 4 + 9 k, 7 + 14 k, 3 + 7 k, 7 + 13 k, -1 - 2 k]] PISOT(18, 324 k + 175) = [[18, 324 k + 175], [9 + 18 k, 7 + 13 k]] PISOT(18, 324 k + 176) = [ 2 [18, 324 k + 176, 5832 k + 6336 k + 1721], [9 + 18 k, 7 + 14 k, 6 + 11 k]] 2 PISOT(18, 324 k + 177) = [[18, 324 k + 177, 5832 k + 6372 k + 1741], [10 + 18 k, -1 - 3 k, -6 - 11 k]] 2 PISOT(18, 324 k + 179) = [[18, 324 k + 179, 5832 k + 6444 k + 1780], [11 + 18 k, -11 - 19 k, 5 + 9 k]] 2 PISOT(18, 324 k + 181) = [[18, 324 k + 181, 5832 k + 6516 k + 1820, 3 2 104976 k + 175932 k + 98281 k + 18301, 4 3 2 1889568 k + 4222368 k + 3538134 k + 1317676 k + 184026], [11 + 18 k, -10 - 17 k, 6 + 9 k, -10 - 17 k, 5 + 9 k]] PISOT(18, 324 k + 182) = [[18, 324 k + 182], [11 + 18 k, -9 - 16 k]] 2 PISOT(18, 324 k + 184) = [[18, 324 k + 184, 5832 k + 6624 k + 1881, 3 2 104976 k + 178848 k + 101572 k + 19229], [10 + 18 k, 3 + 4 k, -7 - 13 k, -4 - 7 k]] 2 PISOT(18, 324 k + 186) = [[18, 324 k + 186, 5832 k + 6696 k + 1922, 3 2 104976 k + 180792 k + 103788 k + 19861], [10 + 18 k, 3 + 6 k, 5 + 8 k, -4 - 7 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.980530974227514 2 PISOT(18, 324 k + 187) = [[18, 324 k + 187, 5832 k + 6732 k + 1943, 3 2 104976 k + 181764 k + 104917 k + 20188, 4 3 2 5 1889568 k + 4362336 k + 3776922 k + 1453450 k + 209756, 34012224 k 4 3 2 + 98152560 k + 113306040 k + 65402767 k + 18876777 k + 2179393], [10 + 18 k, 3 + 7 k, 11 + 19 k, -1 - k, 5 + 8 k, -4 - 7 k]] 2 PISOT(18, 324 k + 189) = [[18, 324 k + 189, 5832 k + 6804 k + 1985, 3 2 104976 k + 183708 k + 107181 k + 20848], [11 + 18 k, -5 - 9 k, -3 - 4 k, 7 + 12 k]] PISOT(18, 324 k + 193) = [[18, 324 k + 193], [11 + 18 k, -3 - 5 k]] 2 PISOT(18, 324 k + 194) = [[18, 324 k + 194, 5832 k + 6984 k + 2091], [12 + 18 k, -14 - 22 k, 9 + 15 k]] 2 PISOT(18, 324 k + 196) = [[18, 324 k + 196, 5832 k + 7056 k + 2134, 3 2 104976 k + 190512 k + 115240 k + 23234, 4 3 2 1889568 k + 4572288 k + 4148712 k + 1672952 k + 252961], [10 + 18 k, 10 + 16 k, -3 - 6 k, -7 - 12 k, -3 - 5 k]] 2 PISOT(18, 324 k + 208) = [[18, 324 k + 208, 5832 k + 7488 k + 2404, 3 2 104976 k + 202176 k + 129808 k + 27785, 4 3 2 1889568 k + 4852224 k + 4672944 k + 2000324 k + 321134], [12 + 18 k, -5 - 8 k, -1 - 2 k, -4 - 5 k, 9 + 14 k]] 2 PISOT(18, 324 k + 212) = [[18, 324 k + 212, 5832 k + 7632 k + 2497, 3 2 104976 k + 206064 k + 134836 k + 29410], [13 + 18 k, -15 - 22 k, 7 + 11 k, 2 + 3 k]] 2 PISOT(18, 324 k + 214) = [[18, 324 k + 214, 5832 k + 7704 k + 2544, 3 2 104976 k + 208008 k + 137380 k + 30243, 4 3 2 1889568 k + 4992192 k + 4945752 k + 2177580 k + 359528], [12 + 18 k, -2 - 2 k, 8 + 12 k, -1 - k, 4 + 6 k]] PISOT(18, 324 k + 215) = 2 [[18, 324 k + 215, 5832 k + 7740 k + 2568], [12 + 18 k, -1 - k, 4 + 6 k]] PISOT(18, 324 k + 217) = 2 [[18, 324 k + 217, 5832 k + 7812 k + 2616], [12 + 18 k, 1 + k, -4 - 6 k]] 2 PISOT(18, 324 k + 218) = [[18, 324 k + 218, 5832 k + 7848 k + 2640, 3 2 104976 k + 211896 k + 142564 k + 31971, 4 3 2 1889568 k + 5085504 k + 5132376 k + 2301996 k + 387176], [12 + 18 k, 2 + 2 k, -8 - 12 k, -1 - k, 4 + 6 k]] 2 PISOT(18, 324 k + 220) = [[18, 324 k + 220, 5832 k + 7920 k + 2689, 3 2 104976 k + 213840 k + 145204 k + 32867, 4 3 2 1889568 k + 5132160 k + 5227308 k + 2366372 k + 401725], [13 + 18 k, -9 - 14 k, -7 - 9 k, 11 + 16 k, -2 - 3 k]] 2 PISOT(18, 324 k + 221) = [[18, 324 k + 221, 5832 k + 7956 k + 2713, 3 2 104976 k + 214812 k + 146509 k + 33305, 4 3 2 5 1889568 k + 5155488 k + 5274450 k + 2398126 k + 408855, 34012224 k 4 3 2 + 115998480 k + 158235768 k + 107920205 k + 36799959 k + 5019139], [13 + 18 k, -9 - 13 k, 2 + 2 k, -8 - 11 k, 6 + 9 k, 2 + 3 k]] PISOT(18, 324 k + 226) = [[18, 324 k + 226], [12 + 18 k, 7 + 10 k]] 2 PISOT(18, 324 k + 232) = [[18, 324 k + 232, 5832 k + 8352 k + 2990, 3 2 104976 k + 225504 k + 161464 k + 38535], [14 + 18 k, -15 - 20 k, 9 + 12 k, -5 - 7 k]] 2 PISOT(18, 324 k + 235) = [[18, 324 k + 235, 5832 k + 8460 k + 3068, 3 2 104976 k + 228420 k + 165673 k + 40054], [13 + 18 k, 1 + k, -3 - 5 k, -8 - 11 k]] 2 PISOT(18, 324 k + 240) = [[18, 324 k + 240, 5832 k + 8640 k + 3200, 3 2 104976 k + 233280 k + 172800 k + 42667, 4 3 2 1889568 k + 5598720 k + 6220800 k + 3072012 k + 568898], [12 + 18 k, 17 + 24 k, 10 + 14 k, 5 + 7 k, 3 + 4 k]] PISOT(18, 324 k + 241) = 2 [[18, 324 k + 241, 5832 k + 8676 k + 3227], [13 + 18 k, 5 + 7 k, 3 + 4 k]] PISOT(18, 324 k + 242) = [[18, 324 k + 242], [13 + 18 k, 6 + 8 k]] PISOT(18, 324 k + 243) = [ 2 [18, 324 k + 243, 5832 k + 8748 k + 3281], [13 + 18 k, 7 + 9 k, -3 - 4 k]] PISOT(18, 324 k + 244) = [[18, 324 k + 244], [14 + 18 k, -6 - 8 k]] 2 PISOT(18, 324 k + 245) = [[18, 324 k + 245, 5832 k + 8820 k + 3335, 3 2 104976 k + 238140 k + 180085 k + 45397], [14 + 18 k, -5 - 7 k, -4 - 5 k, 3 + 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.9718257080 2 PISOT(18, 324 k + 248) = [[18, 324 k + 248, 5832 k + 8928 k + 3417], [14 + 18 k, -4 - 4 k, 13 + 17 k]] 2 PISOT(18, 324 k + 249) = [[18, 324 k + 249, 5832 k + 8964 k + 3445], [14 + 18 k, -3 - 3 k, 10 + 13 k]] 2 PISOT(18, 324 k + 259) = [[18, 324 k + 259, 5832 k + 9324 k + 3727, 3 2 104976 k + 251748 k + 201253 k + 53631, 4 3 2 1889568 k + 6041952 k + 7245018 k + 3861302 k + 771742], [14 + 18 k, 6 + 7 k, -6 - 7 k, 5 + 7 k, 8 + 10 k]] PISOT(18, 324 k + 260) = [[18, 324 k + 260], [15 + 18 k, -8 - 10 k]] 2 PISOT(18, 324 k + 265) = [[18, 324 k + 265, 5832 k + 9540 k + 3901], [14 + 18 k, 10 + 13 k, 9 + 11 k]] 2 PISOT(18, 324 k + 269) = [[18, 324 k + 269, 5832 k + 9684 k + 4020, 3 2 104976 k + 261468 k + 217081 k + 60076], [14 + 18 k, 14 + 17 k, 2 + 2 k, -5 - 6 k]] PISOT(18, 324 k + 277) = [[18, 324 k + 277], [15 + 18 k, 6 + 7 k]] 2 PISOT(18, 324 k + 278) = [[18, 324 k + 278, 5832 k + 10008 k + 4294, 3 2 104976 k + 270216 k + 231868 k + 66325], [16 + 18 k, -8 - 10 k, -9 - 10 k, 6 + 7 k]] 2 PISOT(18, 324 k + 283) = [[18, 324 k + 283, 5832 k + 10188 k + 4449], [17 + 18 k, -21 - 23 k, 14 + 16 k]] 2 PISOT(18, 324 k + 285) = [[18, 324 k + 285, 5832 k + 10260 k + 4513, 3 2 104976 k + 277020 k + 243693 k + 71464], [15 + 18 k, 13 + 15 k, 4 + 4 k, -7 - 8 k]] PISOT(18, 324 k + 295) = [[18, 324 k + 295], [17 + 18 k, -10 - 11 k]] PISOT(18, 324 k + 304) = [[18, 324 k + 304], [16 + 18 k, 15 + 16 k]] PISOT(18, 324 k + 305) = [[18, 324 k + 305], [16 + 18 k, 16 + 17 k]] 2 PISOT(18, 324 k + 313) = [[18, 324 k + 313, 5832 k + 11268 k + 5443, 3 2 104976 k + 304236 k + 293917 k + 94653, 4 3 2 5 1889568 k + 7301664 k + 10580922 k + 6814826 k + 1646002, 34012224 k 4 3 2 + 164287440 k + 317426040 k + 306662185 k + 148135099 k + 28623737], [18 + 18 k, -11 - 11 k, 7 + 7 k, -4 - 4 k, 2 + 2 k, -1 - k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970421624309967 2 PISOT(18, 324 k + 315) = [[18, 324 k + 315, 5832 k + 11340 k + 5513, 3 2 104976 k + 306180 k + 297693 k + 96486, 4 3 2 5 1889568 k + 7348320 k + 10716786 k + 6946686 k + 1688654, 34012224 k 4 3 2 + 165337200 k + 321500664 k + 312592527 k + 151970893 k + 29554053], [17 + 18 k, 9 + 9 k, -4 - 4 k, 2 + 2 k, -1 - k, 1 + k]] 2 PISOT(18, 324 k + 316) = [[18, 324 k + 316, 5832 k + 11376 k + 5548, 3 2 104976 k + 307152 k + 299584 k + 97406, 4 3 2 1889568 k + 7371648 k + 10784880 k + 7012952 k + 1710153], [18 + 18 k, -8 - 8 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(18, 324 k + 317) = [[18, 324 k + 317, 5832 k + 11412 k + 5583, 3 2 104976 k + 308124 k + 301477 k + 98328], [18 + 18 k, -7 - 7 k, 3 + 3 k, -1 - k]] 2 PISOT(18, 324 k + 318) = [[18, 324 k + 318, 5832 k + 11448 k + 5618, 3 2 104976 k + 309096 k + 303372 k + 99251], [18 + 18 k, -6 - 6 k, 2 + 2 k, -1 - k]] PISOT(18, 324 k + 319) = 2 [[18, 324 k + 319, 5832 k + 11484 k + 5653], [18 + 18 k, -5 - 5 k, 1 + k]] PISOT(18, 324 k + 320) = 2 [[18, 324 k + 320, 5832 k + 11520 k + 5689], [18 + 18 k, -4 - 4 k, 1 + k]] PISOT(18, 324 k + 321) = 2 [[18, 324 k + 321, 5832 k + 11556 k + 5725], [18 + 18 k, -3 - 3 k, 1 + k]] PISOT(18, 324 k + 322) = [[18, 324 k + 322], [18 + 18 k, -2 - 2 k]] PISOT(18, 324 k + 323) = [[18, 324 k + 323], [18 + 18 k, -1 - k]] PISOT(19, 361 k + 1) = [[19, 361 k + 1], [19 k, k]] PISOT(19, 361 k + 2) = [[19, 361 k + 2], [19 k, 2 k]] PISOT(19, 361 k + 3) = [[19, 361 k + 3], [19 k, 3 k]] 2 PISOT(19, 361 k + 4) = [[19, 361 k + 4, 6859 k + 152 k + 1], [19 k, 4 k, k]] 2 PISOT(19, 361 k + 5) = [[19, 361 k + 5, 6859 k + 190 k + 1], [19 k, 5 k, k]] PISOT(19, 361 k + 6) = [ 2 3 2 [19, 361 k + 6, 6859 k + 228 k + 2, 130321 k + 6498 k + 112 k + 1], [19 k, 6 k, 2 k, k]] PISOT(19, 361 k + 7) = [ 2 3 2 [19, 361 k + 7, 6859 k + 266 k + 3, 130321 k + 7581 k + 163 k + 1], [19 k, 7 k, 3 k, k]] PISOT(19, 361 k + 8) = [ 2 3 2 [19, 361 k + 8, 6859 k + 304 k + 3, 130321 k + 8664 k + 178 k + 1], [19 k, 8 k, 3 k, k]] 2 PISOT(19, 361 k + 9) = [[19, 361 k + 9, 6859 k + 342 k + 4, 3 2 130321 k + 9747 k + 233 k + 2, 4 3 2 2476099 k + 246924 k + 8949 k + 148 k + 1], [19 k, 9 k, 4 k, 2 k, k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970254721867615 2 PISOT(19, 361 k + 10) = [[19, 361 k + 10, 6859 k + 380 k + 5, 3 2 130321 k + 10830 k + 290 k + 2, 4 3 2 2476099 k + 274360 k + 11115 k + 176 k + 1], [1 + 19 k, -9 k, -5 k, -3 k, -k]] 2 PISOT(19, 361 k + 11) = [[19, 361 k + 11, 6859 k + 418 k + 6, 3 2 130321 k + 11913 k + 349 k + 3, 4 3 2 2476099 k + 301796 k + 13395 k + 246 k + 1], [19 k, 11 k, 6 k, 3 k, k]] PISOT(19, 361 k + 20) = [[19, 361 k + 20], [2 + 19 k, -1 - 18 k]] PISOT(19, 361 k + 21) = [[19, 361 k + 21], [2 + 19 k, -1 - 17 k]] PISOT(19, 361 k + 22) = [[19, 361 k + 22], [2 + 19 k, -1 - 16 k]] PISOT(19, 361 k + 26) = [ 2 3 2 [19, 361 k + 26, 6859 k + 988 k + 36, 130321 k + 28158 k + 2044 k + 50], [1 + 19 k, 7 k, 10 k, 1 + 14 k]] PISOT(19, 361 k + 28) = 2 [[19, 361 k + 28, 6859 k + 1064 k + 41], [1 + 19 k, 9 k, 1 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.977333144991208 2 PISOT(19, 361 k + 33) = [[19, 361 k + 33, 6859 k + 1254 k + 57, 3 2 130321 k + 35739 k + 3255 k + 98, 4 3 2 2476099 k + 905388 k + 123804 k + 7486 k + 168, 5 4 3 2 47045881 k + 21502965 k + 3922626 k + 356505 k + 16101 k + 288], [1 + 19 k, 1 + 14 k, 5 k, 1 + 8 k, -6 k, -1 - 11 k]] PISOT(19, 361 k + 41) = 2 [[19, 361 k + 41, 6859 k + 1558 k + 88], [1 + 19 k, 2 + 22 k, 1 + 9 k]] 2 PISOT(19, 361 k + 43) = [[19, 361 k + 43, 6859 k + 1634 k + 97, 3 2 130321 k + 46569 k + 5535 k + 219, 4 3 2 2476099 k + 1179748 k + 210444 k + 16664 k + 494, 5 4 3 2 47045881 k + 28019015 k + 6666226 k + 792178 k + 47015 k + 1114], [3 + 19 k, -3 - 14 k, 4 + 25 k, -3 - 19 k, 2 + 13 k, -1 - 8 k]] PISOT(19, 361 k + 46) = [[19, 361 k + 46], [2 + 19 k, 1 + 8 k]] PISOT(19, 361 k + 48) = 2 [[19, 361 k + 48, 6859 k + 1824 k + 121], [3 + 19 k, -2 - 9 k, 2 + 15 k]] PISOT(19, 361 k + 52) = [[19, 361 k + 52], [2 + 19 k, 2 + 14 k]] PISOT(19, 361 k + 53) = [[ 2 3 2 19, 361 k + 53, 6859 k + 2014 k + 148, 130321 k + 57399 k + 8433 k + 413 ], [3 + 19 k, -4 k, -2 - 11 k, 1 + 7 k]] PISOT(19, 361 k + 55) = 2 [[19, 361 k + 55, 6859 k + 2090 k + 159], [3 + 19 k, -1 - 2 k, 2 + 13 k]] PISOT(19, 361 k + 59) = 2 [[19, 361 k + 59, 6859 k + 2242 k + 183], [3 + 19 k, 2 k, 1 + 6 k]] PISOT(19, 361 k + 60) = 2 [[19, 361 k + 60, 6859 k + 2280 k + 189], [4 + 19 k, -3 - 16 k, 1 + 6 k]] PISOT(19, 361 k + 61) = 2 [[19, 361 k + 61, 6859 k + 2318 k + 196], [3 + 19 k, 1 + 4 k, -1 - 6 k]] PISOT(19, 361 k + 64) = 2 [[19, 361 k + 64, 6859 k + 2432 k + 216], [4 + 19 k, -3 - 12 k, 3 + 17 k]] PISOT(19, 361 k + 65) = [[19, 361 k + 65], [4 + 19 k, -2 - 11 k]] 2 PISOT(19, 361 k + 66) = [[19, 361 k + 66, 6859 k + 2508 k + 229, 3 2 130321 k + 71478 k + 13058 k + 795], [4 + 19 k, -2 - 10 k, 3 k, 2 + 11 k] ] PISOT(19, 361 k + 71) = [[19, 361 k + 71], [4 + 19 k, -1 - 5 k]] PISOT(19, 361 k + 72) = [[19, 361 k + 72], [3 + 19 k, 3 + 15 k]] 2 PISOT(19, 361 k + 73) = [[19, 361 k + 73, 6859 k + 2774 k + 280, 3 2 130321 k + 79059 k + 15969 k + 1074, 4 3 2 2476099 k + 2002828 k + 606993 k + 81692 k + 4120], [4 + 19 k, -3 k, -3 - 12 k, 2 + 11 k, 1 + 5 k]] 2 PISOT(19, 361 k + 74) = [[19, 361 k + 74, 6859 k + 2812 k + 288, 3 2 130321 k + 80142 k + 16420 k + 1121], [4 + 19 k, -1 - 2 k, 2 + 11 k, 1 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.972567120206416 2 PISOT(19, 361 k + 78) = [[19, 361 k + 78, 6859 k + 2964 k + 320, 3 2 130321 k + 84474 k + 18244 k + 1313], [4 + 19 k, 2 k, 1 + 8 k, 3 + 14 k]] PISOT(19, 361 k + 79) = 2 [[19, 361 k + 79, 6859 k + 3002 k + 328], [5 + 19 k, -4 - 16 k, 2 + 9 k]] 2 PISOT(19, 361 k + 81) = [[19, 361 k + 81, 6859 k + 3078 k + 345, 3 2 130321 k + 87723 k + 19671 k + 1469, 4 3 2 2476099 k + 2222316 k + 747612 k + 111712 k + 6255], [4 + 19 k, 1 + 5 k, 1 + 2 k, -2 - 11 k, -2 - 9 k]] 2 PISOT(19, 361 k + 83) = [[19, 361 k + 83, 6859 k + 3154 k + 363, 3 2 130321 k + 89889 k + 20683 k + 1588, 4 3 2 2476099 k + 2277188 k + 785802 k + 120602 k + 6947, 5 4 3 2 47045881 k + 54083215 k + 24880842 k + 5726297 k + 659363 k + 30391], [4 + 19 k, 2 + 7 k, -2 - 7 k, 2 + 8 k, -3 k, -3 - 13 k]] 2 PISOT(19, 361 k + 84) = [[19, 361 k + 84, 6859 k + 3192 k + 371, 3 2 130321 k + 90972 k + 21154 k + 1639], [4 + 19 k, 2 + 8 k, -3 k, -3 - 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.975627999148667 2 PISOT(19, 361 k + 88) = [[19, 361 k + 88, 6859 k + 3344 k + 408, 3 2 130321 k + 95304 k + 23248 k + 1892, 4 3 2 2476099 k + 2414368 k + 883272 k + 143704 k + 8774], [5 + 19 k, -2 - 7 k, 1 + 6 k, 2 + 9 k, 1 + 4 k]] PISOT(19, 361 k + 89) = 2 [[19, 361 k + 89, 6859 k + 3382 k + 417], [4 + 19 k, 3 + 13 k, 1 + 4 k]] 2 PISOT(19, 361 k + 90) = [[19, 361 k + 90, 6859 k + 3420 k + 426, 3 2 130321 k + 97470 k + 24288 k + 2016], [4 + 19 k, 3 + 14 k, 2 + 9 k, 1 + 4 k]] PISOT(19, 361 k + 91) = [[19, 361 k + 91], [5 + 19 k, -1 - 4 k]] PISOT(19, 361 k + 92) = 2 [[19, 361 k + 92, 6859 k + 3496 k + 445], [5 + 19 k, -1 - 3 k, 1 + 4 k]] PISOT(19, 361 k + 102) = [[19, 361 k + 102], [5 + 19 k, 2 + 7 k]] PISOT(19, 361 k + 108) = [ 2 [19, 361 k + 108, 6859 k + 4104 k + 614], [7 + 19 k, -8 - 25 k, 3 + 10 k]] PISOT(19, 361 k + 109) = [ 2 [19, 361 k + 109, 6859 k + 4142 k + 625], [6 + 19 k, -1 - 5 k, -3 - 10 k]] 2 PISOT(19, 361 k + 111) = [[19, 361 k + 111, 6859 k + 4218 k + 648, 3 2 130321 k + 120213 k + 36945 k + 3783], [6 + 19 k, -1 - 3 k, 1 + k, -4 - 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998711247926498 2 PISOT(19, 361 k + 117) = [[19, 361 k + 117, 6859 k + 4446 k + 720, 3 2 130321 k + 126711 k + 41049 k + 4431], [6 + 19 k, 3 k, 6 + 18 k, -1 - 3 k]] 2 PISOT(19, 361 k + 119) = [[19, 361 k + 119, 6859 k + 4522 k + 745, 3 2 130321 k + 128877 k + 42471 k + 4664], [7 + 19 k, -5 - 14 k, 2 + 7 k, 2 + 6 k]] PISOT(19, 361 k + 120) = [[19, 361 k + 120], [6 + 19 k, 2 + 6 k]] PISOT(19, 361 k + 121) = [[19, 361 k + 121], [7 + 19 k, -4 - 12 k]] 2 PISOT(19, 361 k + 122) = [[19, 361 k + 122, 6859 k + 4636 k + 783, 3 2 130321 k + 132126 k + 44638 k + 5025], [7 + 19 k, -4 - 11 k, 2 + 5 k, -2 - 6 k]] 2 PISOT(19, 361 k + 123) = [[19, 361 k + 123, 6859 k + 4674 k + 796, 3 2 130321 k + 133209 k + 45377 k + 5151, 4 3 2 2476099 k + 3374628 k + 1724421 k + 391554 k + 33333, 5 4 3 2 47045881 k + 80147415 k + 54608470 k + 18600912 k + 3167416 k + 215704] , [7 + 19 k, -3 - 10 k, -3 - 8 k, 2 + 5 k, -2 - 5 k, 2 + 6 k]] 2 PISOT(19, 361 k + 124) = [[19, 361 k + 124, 6859 k + 4712 k + 809, 3 2 130321 k + 134292 k + 46118 k + 5278], [5 + 19 k, 9 + 29 k, 6 + 18 k, 1 + 3 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.971424913066440 2 PISOT(19, 361 k + 125) = [[19, 361 k + 125, 6859 k + 4750 k + 822, 3 2 130321 k + 135375 k + 46861 k + 5405, 4 3 2 2476099 k + 3429500 k + 1780851 k + 410890 k + 35540, 5 4 3 2 47045881 k + 81450625 k + 56396142 k + 19520240 k + 3377454 k + 233689] , [6 + 19 k, 4 + 11 k, -2 - 4 k, 3 + 11 k, 5 + 15 k, 1 + 3 k]] PISOT(19, 361 k + 127) = 2 [[19, 361 k + 127, 6859 k + 4826 k + 849], [7 + 19 k, -3 - 6 k, 6 + 17 k]] PISOT(19, 361 k + 132) = 2 [[19, 361 k + 132, 6859 k + 5016 k + 917], [6 + 19 k, 6 + 18 k, 4 + 11 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.981123249397955 2 PISOT(19, 361 k + 134) = [[19, 361 k + 134, 6859 k + 5092 k + 945, 3 2 130321 k + 145122 k + 53866 k + 6664, 4 3 2 2476099 k + 3676424 k + 2046927 k + 506492 k + 46994, 5 4 3 2 47045881 k + 87315070 k + 64819716 k + 24059036 k + 4464749 k + 331398] , [6 + 19 k, 6 + 20 k, 9 + 27 k, 6 + 19 k, 7 + 20 k, 3 + 8 k]] PISOT(19, 361 k + 139) = [[19, 361 k + 139], [8 + 19 k, -5 - 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.997367307066144 2 PISOT(19, 361 k + 140) = [[19, 361 k + 140, 6859 k + 5320 k + 1032, 3 2 130321 k + 151620 k + 58816 k + 7607], [7 + 19 k, 3 + 7 k, -1 - 5 k, -7 - 18 k]] PISOT(19, 361 k + 144) = [ 2 [19, 361 k + 144, 6859 k + 5472 k + 1091], [8 + 19 k, -4 - 8 k, 6 + 15 k]] PISOT(19, 361 k + 146) = 2 [[19, 361 k + 146, 6859 k + 5548 k + 1122], [7 + 19 k, 5 + 13 k, 2 + 5 k]] 2 PISOT(19, 361 k + 147) = [[19, 361 k + 147, 6859 k + 5586 k + 1137, 3 2 130321 k + 159201 k + 64815 k + 8794], [7 + 19 k, 5 + 14 k, 5 + 13 k, 2 + 5 k]] PISOT(19, 361 k + 151) = [ 2 [19, 361 k + 151, 6859 k + 5738 k + 1200], [9 + 19 k, -9 - 20 k, 5 + 12 k] ] PISOT(19, 361 k + 154) = [ 2 [19, 361 k + 154, 6859 k + 5852 k + 1248], [9 + 19 k, -8 - 17 k, 6 + 14 k] ] PISOT(19, 361 k + 155) = [ 2 [19, 361 k + 155, 6859 k + 5890 k + 1264], [8 + 19 k, 2 + 3 k, -6 - 14 k]] PISOT(19, 361 k + 159) = [ 2 [19, 361 k + 159, 6859 k + 6042 k + 1331], [7 + 19 k, 11 + 26 k, 4 + 9 k]] PISOT(19, 361 k + 160) = 2 [[19, 361 k + 160, 6859 k + 6080 k + 1347], [8 + 19 k, 4 + 8 k, -4 - 9 k]] PISOT(19, 361 k + 161) = [[19, 361 k + 161], [8 + 19 k, 4 + 9 k]] 2 PISOT(19, 361 k + 162) = [[19, 361 k + 162, 6859 k + 6156 k + 1381, 3 2 130321 k + 175446 k + 78722 k + 11773], [8 + 19 k, 5 + 10 k, -4 - 10 k, -4 - 9 k]] 2 PISOT(19, 361 k + 163) = [[19, 361 k + 163, 6859 k + 6194 k + 1398, 3 2 130321 k + 176529 k + 79693 k + 11990], [9 + 19 k, -3 - 8 k, -6 - 12 k, 5 + 11 k]] 2 PISOT(19, 361 k + 164) = [[19, 361 k + 164, 6859 k + 6232 k + 1416, 3 2 130321 k + 177612 k + 80704 k + 12226], [10 + 19 k, -13 - 26 k, 11 + 23 k, -5 - 11 k]] PISOT(19, 361 k + 166) = [ 2 [19, 361 k + 166, 6859 k + 6308 k + 1450], [9 + 19 k, -3 - 5 k, 6 + 13 k]] 2 PISOT(19, 361 k + 174) = [[19, 361 k + 174, 6859 k + 6612 k + 1593, 3 2 130321 k + 188442 k + 90810 k + 14584, 4 3 2 2476099 k + 4773864 k + 3450951 k + 1108556 k + 133517], [9 + 19 k, 2 + 3 k, -5 - 11 k, -3 - 6 k, 1 + 2 k]] 2 PISOT(19, 361 k + 176) = [[19, 361 k + 176, 6859 k + 6688 k + 1630, 3 2 130321 k + 190608 k + 92916 k + 15096], [9 + 19 k, 2 + 5 k, 4 + 8 k, -1 - 2 k]] 2 PISOT(19, 361 k + 177) = [[19, 361 k + 177, 6859 k + 6726 k + 1649, 3 2 130321 k + 191691 k + 93991 k + 15363], [10 + 19 k, -7 - 13 k, 6 + 12 k, -1 - 2 k]] PISOT(19, 361 k + 180) = [[19, 361 k + 180], [10 + 19 k, -5 - 10 k]] PISOT(19, 361 k + 181) = [[19, 361 k + 181], [9 + 19 k, 5 + 10 k]] 2 PISOT(19, 361 k + 184) = [[19, 361 k + 184, 6859 k + 6992 k + 1782, 3 2 130321 k + 199272 k + 101572 k + 17258], [9 + 19 k, 6 + 13 k, 6 + 12 k, 1 + 2 k]] 2 PISOT(19, 361 k + 185) = [[19, 361 k + 185, 6859 k + 7030 k + 1801, 3 2 130321 k + 200355 k + 102663 k + 17533], [10 + 19 k, -3 - 5 k, 4 + 8 k, 1 + 2 k]] 2 PISOT(19, 361 k + 187) = [[19, 361 k + 187, 6859 k + 7106 k + 1840, 3 2 130321 k + 202521 k + 104889 k + 18105, 4 3 2 2476099 k + 5130532 k + 3985953 k + 1376150 k + 178147], [10 + 19 k, -1 - 3 k, -6 - 11 k, 3 + 6 k, 1 + 2 k]] PISOT(19, 361 k + 195) = [ 2 [19, 361 k + 195, 6859 k + 7410 k + 2001], [10 + 19 k, 2 + 5 k, 7 + 13 k]] 2 PISOT(19, 361 k + 197) = [[19, 361 k + 197, 6859 k + 7486 k + 2043, 3 2 130321 k + 213351 k + 116443 k + 21187], [9 + 19 k, 13 + 26 k, 12 + 23 k, 6 + 11 k]] 2 PISOT(19, 361 k + 198) = [[19, 361 k + 198, 6859 k + 7524 k + 2063, 3 2 130321 k + 214434 k + 117598 k + 21495], [10 + 19 k, 5 + 8 k, -6 - 12 k, -6 - 11 k]] 2 PISOT(19, 361 k + 199) = [[19, 361 k + 199, 6859 k + 7562 k + 2084, 3 2 130321 k + 215517 k + 118793 k + 21824], [11 + 19 k, -5 - 10 k, -6 - 10 k, 5 + 9 k]] PISOT(19, 361 k + 200) = [[19, 361 k + 200], [11 + 19 k, -5 - 9 k]] PISOT(19, 361 k + 201) = [ 2 [19, 361 k + 201, 6859 k + 7638 k + 2126], [11 + 19 k, -4 - 8 k, -5 - 9 k] ] 2 PISOT(19, 361 k + 202) = [[19, 361 k + 202, 6859 k + 7676 k + 2148], [12 + 19 k, -15 - 26 k, 5 + 9 k]] 2 PISOT(19, 361 k + 206) = [[19, 361 k + 206, 6859 k + 7828 k + 2233], [11 + 19 k, -1 - 3 k, -8 - 14 k]] PISOT(19, 361 k + 207) = [ 2 [19, 361 k + 207, 6859 k + 7866 k + 2255], [10 + 19 k, 9 + 17 k, 8 + 14 k] ] 2 PISOT(19, 361 k + 210) = [[19, 361 k + 210, 6859 k + 7980 k + 2321], [10 + 19 k, 11 + 20 k, 7 + 12 k]] 2 PISOT(19, 361 k + 214) = [[19, 361 k + 214, 6859 k + 8132 k + 2410, 3 2 130321 k + 231762 k + 137376 k + 27141], [12 + 19 k, -9 - 14 k, 8 + 13 k, -3 - 5 k]] PISOT(19, 361 k + 215) = [ 2 [19, 361 k + 215, 6859 k + 8170 k + 2433], [12 + 19 k, -8 - 13 k, 3 + 5 k] ] PISOT(19, 361 k + 217) = [ 2 [19, 361 k + 217, 6859 k + 8246 k + 2478], [11 + 19 k, 4 + 8 k, 9 + 15 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.997366358805359 2 PISOT(19, 361 k + 221) = [[19, 361 k + 221, 6859 k + 8398 k + 2571, 3 2 130321 k + 239343 k + 146539 k + 29910], [12 + 19 k, -4 - 7 k, -4 - 5 k, 11 + 18 k]] PISOT(19, 361 k + 222) = [[19, 361 k + 222], [11 + 19 k, 8 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.981123544426591 2 PISOT(19, 361 k + 227) = [[19, 361 k + 227, 6859 k + 8626 k + 2712, 3 2 130321 k + 245841 k + 154585 k + 32401, 4 3 2 5 2476099 k + 6227972 k + 5874249 k + 2462486 k + 387104, 47045881 k 4 3 2 + 147914335 k + 186018246 k + 116968502 k + 36774950 k + 4624842], [13 + 19 k, -14 - 20 k, 18 + 27 k, -13 - 19 k, 13 + 20 k, -5 - 8 k]] 2 PISOT(19, 361 k + 229) = [[19, 361 k + 229, 6859 k + 8702 k + 2760], [13 + 19 k, -12 - 18 k, 7 + 11 k]] PISOT(19, 361 k + 234) = [ 2 [19, 361 k + 234, 6859 k + 8892 k + 2882], [12 + 19 k, 3 + 6 k, 11 + 17 k] ] Take it with a grain of salt, for k=1000 the Pisot index is, 0.971426148905127 2 PISOT(19, 361 k + 236) = [[19, 361 k + 236, 6859 k + 8968 k + 2931, 3 2 130321 k + 255588 k + 167074 k + 36402, 4 3 2 5 2476099 k + 6474896 k + 6348945 k + 2766708 k + 452100, 47045881 k 4 3 2 + 153778780 k + 201052452 k + 131423246 k + 42952305 k + 5614923], [13 + 19 k, -7 - 11 k, -2 - 4 k, -8 - 11 k, 10 + 15 k, -2 - 3 k]] 2 PISOT(19, 361 k + 237) = [[19, 361 k + 237, 6859 k + 9006 k + 2956, 3 2 130321 k + 256671 k + 168497 k + 36869], [14 + 19 k, -20 - 29 k, 12 + 18 k, -2 - 3 k]] 2 PISOT(19, 361 k + 238) = [[19, 361 k + 238, 6859 k + 9044 k + 2981, 3 2 130321 k + 257754 k + 169922 k + 37338, 4 3 2 5 2476099 k + 6529768 k + 6457131 k + 2837800 k + 467671, 47045881 k 4 3 2 + 155081990 k + 204477620 k + 134798818 k + 44430747 k + 5857736], [12 + 19 k, 7 + 10 k, -5 - 8 k, -3 - 5 k, -3 - 5 k, -4 - 6 k]] 2 PISOT(19, 361 k + 239) = [[19, 361 k + 239, 6859 k + 9082 k + 3006, 3 2 130321 k + 258837 k + 171349 k + 37808], [12 + 19 k, 7 + 11 k, 3 + 5 k, 4 + 6 k]] PISOT(19, 361 k + 240) = [[19, 361 k + 240], [12 + 19 k, 8 + 12 k]] PISOT(19, 361 k + 241) = [[19, 361 k + 241], [13 + 19 k, -4 - 6 k]] 2 PISOT(19, 361 k + 242) = [[19, 361 k + 242, 6859 k + 9196 k + 3082, 3 2 130321 k + 262086 k + 175680 k + 39251], [12 + 19 k, 9 + 14 k, 5 + 7 k, -4 - 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.998710388662175 2 PISOT(19, 361 k + 244) = [[19, 361 k + 244, 6859 k + 9272 k + 3133, 3 2 130321 k + 264252 k + 178590 k + 40228], [13 + 19 k, -3 - 3 k, 12 + 18 k, 2 + 3 k]] 2 PISOT(19, 361 k + 250) = [[19, 361 k + 250, 6859 k + 9500 k + 3289, 3 2 130321 k + 270750 k + 187482 k + 43270], [13 + 19 k, 2 + 3 k, k, 9 + 13 k]] PISOT(19, 361 k + 252) = [ 2 [19, 361 k + 252, 6859 k + 9576 k + 3342], [13 + 19 k, 4 + 5 k, -7 - 10 k] ] 2 PISOT(19, 361 k + 253) = [[19, 361 k + 253, 6859 k + 9614 k + 3369], [12 + 19 k, 17 + 25 k, 7 + 10 k]] PISOT(19, 361 k + 259) = [[19, 361 k + 259], [14 + 19 k, -5 - 7 k]] PISOT(19, 361 k + 269) = [ 2 [19, 361 k + 269, 6859 k + 10222 k + 3808], [14 + 19 k, 2 + 3 k, 3 + 4 k]] PISOT(19, 361 k + 270) = [[19, 361 k + 270], [14 + 19 k, 3 + 4 k]] 2 PISOT(19, 361 k + 271) = [[19, 361 k + 271, 6859 k + 10298 k + 3865, 3 2 130321 k + 293493 k + 220311 k + 55123], [15 + 19 k, -11 - 14 k, 7 + 9 k, -3 - 4 k]] 2 PISOT(19, 361 k + 272) = [[19, 361 k + 272, 6859 k + 10336 k + 3894], [15 + 19 k, -10 - 13 k, 3 + 4 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.975626817576613 2 PISOT(19, 361 k + 273) = [[19, 361 k + 273, 6859 k + 10374 k + 3923, 3 2 130321 k + 295659 k + 223603 k + 56373, 4 3 2 2476099 k + 7490028 k + 8496762 k + 4284132 k + 810073], [14 + 19 k, 5 + 7 k, 5 + 6 k, -7 - 9 k, 3 + 4 k]] 2 PISOT(19, 361 k + 277) = [[19, 361 k + 277, 6859 k + 10526 k + 4038, 3 2 130321 k + 299991 k + 230173 k + 58864], [15 + 19 k, -6 - 8 k, -3 - 3 k, 10 + 13 k]] 2 PISOT(19, 361 k + 278) = [[19, 361 k + 278, 6859 k + 10564 k + 4068, 3 2 130321 k + 301074 k + 231868 k + 59527, 4 3 2 5 2476099 k + 7627208 k + 8810832 k + 4523834 k + 871058, 47045881 k 4 3 2 + 181146190 k + 279006792 k + 214875749 k + 82745840 k + 12746183], [15 + 19 k, -5 - 7 k, -5 - 7 k, -6 - 8 k, -3 - 3 k, 10 + 13 k]] 2 PISOT(19, 361 k + 280) = [[19, 361 k + 280, 6859 k + 10640 k + 4126, 3 2 130321 k + 303240 k + 235188 k + 60800, 4 3 2 2476099 k + 7682080 k + 8937258 k + 4620960 k + 895938], [15 + 19 k, -4 - 5 k, 1 + 2 k, 9 + 11 k, -7 - 9 k]] 2 PISOT(19, 361 k + 282) = [[19, 361 k + 282, 6859 k + 10716 k + 4185], [14 + 19 k, 12 + 16 k, 7 + 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.972567645919480 2 PISOT(19, 361 k + 283) = [[19, 361 k + 283, 6859 k + 10754 k + 4215, 3 2 130321 k + 306489 k + 240259 k + 62778], [15 + 19 k, -2 - 2 k, 7 + 8 k, -11 - 14 k]] 2 PISOT(19, 361 k + 287) = [[19, 361 k + 287, 6859 k + 10906 k + 4335, 3 2 130321 k + 310821 k + 247099 k + 65478], [15 + 19 k, 1 + 2 k, 9 + 11 k, -4 - 5 k]] 2 PISOT(19, 361 k + 288) = [[19, 361 k + 288, 6859 k + 10944 k + 4365, 3 2 130321 k + 311904 k + 248814 k + 66157, 4 3 2 2476099 k + 7901568 k + 9455103 k + 5028206 k + 1002692], [15 + 19 k, 3 + 3 k, -9 - 12 k, -9 - 11 k, 4 + 5 k]] PISOT(19, 361 k + 289) = [[19, 361 k + 289], [16 + 19 k, -12 - 15 k]] PISOT(19, 361 k + 290) = [[19, 361 k + 290], [15 + 19 k, 4 + 5 k]] 2 PISOT(19, 361 k + 295) = [[19, 361 k + 295, 6859 k + 11210 k + 4580, 3 2 130321 k + 319485 k + 261065 k + 71106], [15 + 19 k, 8 + 10 k, 3 + 3 k, -9 - 11 k]] PISOT(19, 361 k + 296) = [[19, 361 k + 296], [15 + 19 k, 9 + 11 k]] 2 PISOT(19, 361 k + 297) = [[19, 361 k + 297, 6859 k + 11286 k + 4643], [15 + 19 k, 9 + 12 k, 14 + 17 k]] 2 PISOT(19, 361 k + 300) = [[19, 361 k + 300, 6859 k + 11400 k + 4737], [16 + 19 k, -3 - 4 k, -5 - 6 k]] 2 PISOT(19, 361 k + 301) = [[19, 361 k + 301, 6859 k + 11438 k + 4768], [15 + 19 k, 13 + 16 k, 5 + 6 k]] PISOT(19, 361 k + 302) = [ 2 [19, 361 k + 302, 6859 k + 11476 k + 4800], [16 + 19 k, -2 - 2 k, 5 + 6 k] ] 2 PISOT(19, 361 k + 306) = [[19, 361 k + 306, 6859 k + 11628 k + 4928], [16 + 19 k, 1 + 2 k, 11 + 13 k]] 2 PISOT(19, 361 k + 308) = [[19, 361 k + 308, 6859 k + 11704 k + 4993, 3 2 130321 k + 333564 k + 284598 k + 80942], [16 + 19 k, 4 + 4 k, -9 - 11 k, -6 - 7 k]] PISOT(19, 361 k + 309) = [[19, 361 k + 309], [17 + 19 k, -12 - 14 k]] 2 PISOT(19, 361 k + 313) = [[19, 361 k + 313, 6859 k + 11894 k + 5156], [16 + 19 k, 7 + 9 k, 13 + 15 k]] PISOT(19, 361 k + 315) = [[19, 361 k + 315], [17 + 19 k, -7 - 8 k]] 2 PISOT(19, 361 k + 318) = [[19, 361 k + 318, 6859 k + 12084 k + 5322, 3 2 130321 k + 344394 k + 303360 k + 89068, 4 3 2 5 2476099 k + 8724648 k + 11527794 k + 6769376 k + 1490625, 47045881 k 4 3 2 + 207210390 k + 365048976 k + 321551220 k + 141614682 k + 24946815], [16 + 19 k, 11 + 14 k, 21 + 25 k, 16 + 19 k, 11 + 13 k, 7 + 8 k]] 2 PISOT(19, 361 k + 320) = [[19, 361 k + 320, 6859 k + 12160 k + 5389], [18 + 19 k, -20 - 22 k, 8 + 9 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.977334117747260 2 PISOT(19, 361 k + 328) = [[19, 361 k + 328, 6859 k + 12464 k + 5662, 3 2 130321 k + 355224 k + 322740 k + 97739, 4 3 2 5 2476099 k + 8999008 k + 12264234 k + 7428354 k + 1687197, 47045881 k 4 3 2 + 213726440 k + 388369576 k + 352852393 k + 160288514 k + 29124850], [18 + 19 k, -13 - 14 k, 5 + 5 k, -7 - 8 k, -6 - 6 k, 10 + 11 k]] 2 PISOT(19, 361 k + 333) = [[19, 361 k + 333, 6859 k + 12654 k + 5836], [18 + 19 k, -9 - 9 k, 12 + 13 k]] 2 PISOT(19, 361 k + 334) = [[19, 361 k + 334, 6859 k + 12692 k + 5871, 3 2 130321 k + 361722 k + 334654 k + 103200, 4 3 2 2476099 k + 9163624 k + 12716985 k + 7843428 k + 1814042], [17 + 19 k, 10 + 11 k, 3 + 3 k, -3 - 4 k, -12 - 13 k]] 2 PISOT(19, 361 k + 335) = [[19, 361 k + 335, 6859 k + 12730 k + 5907, 3 2 130321 k + 362805 k + 336691 k + 104157], [18 + 19 k, -7 - 7 k, 10 + 10 k, -13 - 14 k]] PISOT(19, 361 k + 339) = [[19, 361 k + 339], [17 + 19 k, 15 + 16 k]] PISOT(19, 361 k + 340) = [[19, 361 k + 340], [17 + 19 k, 16 + 17 k]] PISOT(19, 361 k + 341) = [[19, 361 k + 341], [17 + 19 k, 17 + 18 k]] 2 PISOT(19, 361 k + 350) = [[19, 361 k + 350, 6859 k + 13300 k + 6447, 3 2 130321 k + 379050 k + 367486 k + 118754, 4 3 2 2476099 k + 9602600 k + 13964601 k + 9025552 k + 2187453], [19 + 19 k, -11 - 11 k, 6 + 6 k, -3 - 3 k, 1 + k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.970256405511744 2 PISOT(19, 361 k + 351) = [[19, 361 k + 351, 6859 k + 13338 k + 6484, 3 2 130321 k + 380133 k + 369593 k + 119779, 4 3 2 2476099 k + 9630036 k + 14044629 k + 9103370 k + 2212679], [18 + 19 k, 9 + 9 k, -5 - 5 k, 3 + 3 k, -1 - k]] 2 PISOT(19, 361 k + 352) = [[19, 361 k + 352, 6859 k + 13376 k + 6521, 3 2 130321 k + 381216 k + 371702 k + 120805, 4 3 2 2476099 k + 9657472 k + 14124771 k + 9181374 k + 2237977], [19 + 19 k, -9 - 9 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(19, 361 k + 353) = [[19, 361 k + 353, 6859 k + 13414 k + 6558, 3 2 130321 k + 382299 k + 373813 k + 121834], [19 + 19 k, -8 - 8 k, 3 + 3 k, -1 - k]] 2 PISOT(19, 361 k + 354) = [[19, 361 k + 354, 6859 k + 13452 k + 6596, 3 2 130321 k + 383382 k + 375964 k + 122902], [19 + 19 k, -7 - 7 k, 3 + 3 k, -1 - k]] 2 PISOT(19, 361 k + 355) = [[19, 361 k + 355, 6859 k + 13490 k + 6633, 3 2 130321 k + 384465 k + 378079 k + 123934], [19 + 19 k, -6 - 6 k, 2 + 2 k, -1 - k]] PISOT(19, 361 k + 356) = 2 [[19, 361 k + 356, 6859 k + 13528 k + 6670], [19 + 19 k, -5 - 5 k, 1 + k]] PISOT(19, 361 k + 357) = 2 [[19, 361 k + 357, 6859 k + 13566 k + 6708], [19 + 19 k, -4 - 4 k, 1 + k]] PISOT(19, 361 k + 358) = [[19, 361 k + 358], [19 + 19 k, -3 - 3 k]] PISOT(19, 361 k + 359) = [[19, 361 k + 359], [19 + 19 k, -2 - 2 k]] PISOT(19, 361 k + 360) = [[19, 361 k + 360], [19 + 19 k, -1 - k]] PISOT(20, 400 k + 1) = [[20, 400 k + 1], [20 k, k]] PISOT(20, 400 k + 2) = [[20, 400 k + 2], [20 k, 2 k]] PISOT(20, 400 k + 3) = [[20, 400 k + 3], [20 k, 3 k]] 2 PISOT(20, 400 k + 4) = [[20, 400 k + 4, 8000 k + 160 k + 1], [20 k, 4 k, k]] 2 PISOT(20, 400 k + 5) = [[20, 400 k + 5, 8000 k + 200 k + 1], [20 k, 5 k, k]] PISOT(20, 400 k + 6) = [ 2 3 2 [20, 400 k + 6, 8000 k + 240 k + 2, 160000 k + 7200 k + 116 k + 1], [20 k, 6 k, 2 k, k]] PISOT(20, 400 k + 7) = [ 2 3 2 [20, 400 k + 7, 8000 k + 280 k + 2, 160000 k + 8400 k + 129 k + 1], [20 k, 7 k, 2 k, k]] PISOT(20, 400 k + 8) = [ 2 3 2 [20, 400 k + 8, 8000 k + 320 k + 3, 160000 k + 9600 k + 184 k + 1], [20 k, 8 k, 3 k, k]] 2 PISOT(20, 400 k + 9) = [[20, 400 k + 9, 8000 k + 360 k + 4, 3 2 160000 k + 10800 k + 241 k + 2, 4 3 2 3200000 k + 288000 k + 9660 k + 152 k + 1], [20 k, 9 k, 4 k, 2 k, k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.972801372680209 2 PISOT(20, 400 k + 10) = [[20, 400 k + 10, 8000 k + 400 k + 5, 3 2 160000 k + 12000 k + 300 k + 3, 4 3 2 3200000 k + 320000 k + 12000 k + 220 k + 2, 5 4 3 2 64000000 k + 8000000 k + 400000 k + 10600 k + 165 k + 1], [1 + 20 k, -10 k, -5 k, -2 k, -k, -k]] 2 PISOT(20, 400 k + 11) = [[20, 400 k + 11, 8000 k + 440 k + 6, 3 2 160000 k + 13200 k + 361 k + 3, 4 3 2 3200000 k + 352000 k + 14460 k + 252 k + 1], [20 k, 11 k, 6 k, 3 k, k]] 2 PISOT(20, 400 k + 12) = [[20, 400 k + 12, 8000 k + 480 k + 7, 3 2 160000 k + 14400 k + 424 k + 4, 4 3 2 3200000 k + 384000 k + 17040 k + 328 k + 2, 5 4 3 2 64000000 k + 9600000 k + 569600 k + 16608 k + 225 k + 1], [20 k, 12 k, 7 k, 4 k, 2 k, k]] PISOT(20, 400 k + 21) = [[20, 400 k + 21], [2 + 20 k, -1 - 19 k]] PISOT(20, 400 k + 22) = [[20, 400 k + 22], [2 + 20 k, -1 - 18 k]] PISOT(20, 400 k + 23) = [[20, 400 k + 23], [2 + 20 k, -1 - 17 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.983851245585674 PISOT(20, 400 k + 27) = [ 2 3 2 [20, 400 k + 27, 8000 k + 1080 k + 36, 160000 k + 32400 k + 2169 k + 48] , [1 + 20 k, 1 + 7 k, -11 k, -1 - 15 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.980902997488959 2 PISOT(20, 400 k + 28) = [[20, 400 k + 28, 8000 k + 1120 k + 39, 3 2 160000 k + 33600 k + 2344 k + 54, 4 3 2 3200000 k + 896000 k + 93840 k + 4344 k + 75], [2 + 20 k, -1 - 12 k, 3 k, 1 + 4 k, -1 - 14 k]] PISOT(20, 400 k + 33) = 2 [[20, 400 k + 33, 8000 k + 1320 k + 54], [2 + 20 k, -7 k, -1 - 12 k]] 2 PISOT(20, 400 k + 34) = [[20, 400 k + 34, 8000 k + 1360 k + 58, 3 2 160000 k + 40800 k + 3476 k + 99, 4 3 2 3200000 k + 1088000 k + 138960 k + 7904 k + 169], [1 + 20 k, 1 + 14 k, 4 k, 7 k, 1 + 12 k]] 2 PISOT(20, 400 k + 36) = [[20, 400 k + 36, 8000 k + 1440 k + 65, 3 2 160000 k + 43200 k + 3896 k + 117, 4 3 2 3200000 k + 1152000 k + 155760 k + 9360 k + 211, 5 4 3 2 64000000 k + 28800000 k + 5190400 k + 467856 k + 21089 k + 381], [2 + 20 k, -1 - 4 k, 1 + 13 k, 3 k, 6 k, 1 + 11 k]] PISOT(20, 400 k + 37) = 2 [[20, 400 k + 37, 8000 k + 1480 k + 68], [1 + 20 k, 1 + 17 k, 1 + 11 k]] 2 PISOT(20, 400 k + 41) = [[20, 400 k + 41, 8000 k + 1640 k + 84, 3 2 160000 k + 49200 k + 5041 k + 172, 4 3 2 3200000 k + 1312000 k + 201660 k + 13768 k + 352], [2 + 20 k, -1 + k, 2 + 22 k, 5 k, 1 + 10 k]] PISOT(20, 400 k + 44) = 2 [[20, 400 k + 44, 8000 k + 1760 k + 97], [2 + 20 k, 4 k, 1 + 9 k]] PISOT(20, 400 k + 45) = 2 [[20, 400 k + 45, 8000 k + 1800 k + 101], [2 + 20 k, 1 + 5 k, -1 - 9 k]] PISOT(20, 400 k + 49) = [[ 2 3 2 20, 400 k + 49, 8000 k + 1960 k + 120, 160000 k + 58800 k + 7201 k + 294 ], [3 + 20 k, -2 - 11 k, 2 + 13 k, -1 - 8 k]] PISOT(20, 400 k + 50) = [[ 2 3 2 20, 400 k + 50, 8000 k + 2000 k + 125, 160000 k + 60000 k + 7500 k + 313 ], [3 + 20 k, -1 - 10 k, -1 - 5 k, 1 + 8 k]] PISOT(20, 400 k + 51) = 2 [[20, 400 k + 51, 8000 k + 2040 k + 130], [2 + 20 k, 1 + 11 k, 1 + 8 k]] PISOT(20, 400 k + 57) = 2 [[20, 400 k + 57, 8000 k + 2280 k + 162], [4 + 20 k, -4 - 23 k, 2 + 14 k]] 2 PISOT(20, 400 k + 61) = [[20, 400 k + 61, 8000 k + 2440 k + 186, 3 2 160000 k + 73200 k + 11161 k + 567], [4 + 20 k, -4 - 19 k, 4 + 22 k, -2 - 13 k]] PISOT(20, 400 k + 66) = [[20, 400 k + 66], [3 + 20 k, 1 + 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.989005296549418 2 PISOT(20, 400 k + 67) = [[20, 400 k + 67, 8000 k + 2680 k + 224, 3 2 160000 k + 80400 k + 13449 k + 749, 4 3 2 3200000 k + 2144000 k + 538140 k + 59976 k + 2504], [3 + 20 k, 1 + 7 k, 1 + 3 k, -2 - 10 k, 1 + 6 k]] 2 PISOT(20, 400 k + 68) = [[20, 400 k + 68, 8000 k + 2720 k + 231, 3 2 160000 k + 81600 k + 13864 k + 785, 4 3 2 3200000 k + 2176000 k + 554640 k + 62816 k + 2668], [3 + 20 k, 1 + 8 k, 1 + 7 k, 1 + 4 k, -1 - 6 k]] PISOT(20, 400 k + 72) = 2 [[20, 400 k + 72, 8000 k + 2880 k + 259], [4 + 20 k, -2 - 8 k, 2 + 11 k]] PISOT(20, 400 k + 75) = 2 [[20, 400 k + 75, 8000 k + 3000 k + 281], [3 + 20 k, 2 + 15 k, 3 + 16 k]] 2 PISOT(20, 400 k + 77) = [[20, 400 k + 77, 8000 k + 3080 k + 296, 3 2 160000 k + 92400 k + 17769 k + 1138, 4 3 2 3200000 k + 2464000 k + 710940 k + 91104 k + 4375], [4 + 20 k, -1 - 3 k, 2 + 8 k, -2 - 9 k, 1 + 5 k]] 2 PISOT(20, 400 k + 78) = [[20, 400 k + 78, 8000 k + 3120 k + 304, 3 2 160000 k + 93600 k + 18244 k + 1185], [5 + 20 k, -5 - 22 k, 3 + 14 k, -1 - 5 k]] PISOT(20, 400 k + 81) = 2 [[20, 400 k + 81, 8000 k + 3240 k + 328], [3 + 20 k, 4 + 21 k, 1 + 5 k]] 2 PISOT(20, 400 k + 82) = [[20, 400 k + 82, 8000 k + 3280 k + 336, 3 2 160000 k + 98400 k + 20164 k + 1377], [5 + 20 k, -4 - 18 k, 1 + 6 k, 1 + 5 k]] PISOT(20, 400 k + 86) = [[20, 400 k + 86], [5 + 20 k, -3 - 14 k]] 2 PISOT(20, 400 k + 88) = [[20, 400 k + 88, 8000 k + 3520 k + 387, 3 2 160000 k + 105600 k + 23224 k + 1702], [5 + 20 k, -3 - 12 k, 2 + 7 k, -2 - 9 k]] PISOT(20, 400 k + 89) = [[20, 400 k + 89], [4 + 20 k, 2 + 9 k]] PISOT(20, 400 k + 93) = [[20, 400 k + 93], [4 + 20 k, 3 + 13 k]] 2 PISOT(20, 400 k + 98) = [[20, 400 k + 98, 8000 k + 3920 k + 480, 3 2 160000 k + 117600 k + 28804 k + 2351, 4 3 2 3200000 k + 3136000 k + 1152240 k + 188120 k + 11515], [5 + 20 k, -1 - 2 k, 2 + 10 k, 2 + 9 k, 1 + 4 k]] PISOT(20, 400 k + 101) = 2 [[20, 400 k + 101, 8000 k + 4040 k + 510], [6 + 20 k, -5 - 19 k, 1 + 4 k]] 2 PISOT(20, 400 k + 102) = [[20, 400 k + 102, 8000 k + 4080 k + 520, 3 2 160000 k + 122400 k + 31204 k + 2651, 4 3 2 3200000 k + 3264000 k + 1248240 k + 212120 k + 13515], [5 + 20 k, 1 + 2 k, -2 - 10 k, -3 - 11 k, 1 + 4 k]] PISOT(20, 400 k + 109) = [[20, 400 k + 109], [6 + 20 k, -3 - 11 k]] 2 PISOT(20, 400 k + 110) = [[20, 400 k + 110, 8000 k + 4400 k + 605, 3 2 160000 k + 132000 k + 36300 k + 3328, 4 3 2 3200000 k + 3520000 k + 1452000 k + 266220 k + 18307, 5 4 3 2 64000000 k + 88000000 k + 48400000 k + 13310600 k + 1830465 k + 100705] , [5 + 20 k, 3 + 10 k, -1 - 5 k, -2 - 7 k, 2 k, 3 + 11 k]] PISOT(20, 400 k + 113) = [[20, 400 k + 113], [6 + 20 k, -2 - 7 k]] PISOT(20, 400 k + 114) = [[20, 400 k + 114], [5 + 20 k, 4 + 14 k]] 2 PISOT(20, 400 k + 119) = [[20, 400 k + 119, 8000 k + 4760 k + 708, 3 2 160000 k + 142800 k + 42481 k + 4212], [7 + 20 k, -6 - 21 k, -2 - 5 k, 3 + 10 k]] PISOT(20, 400 k + 125) = 2 [[20, 400 k + 125, 8000 k + 5000 k + 781], [5 + 20 k, 7 + 25 k, 5 + 16 k]] 2 PISOT(20, 400 k + 132) = [[20, 400 k + 132, 8000 k + 5280 k + 871, 3 2 160000 k + 158400 k + 52264 k + 5747], [7 + 20 k, -3 - 8 k, 2 + 7 k, 2 + 6 k]] PISOT(20, 400 k + 133) = 2 [[20, 400 k + 133, 8000 k + 5320 k + 884], [6 + 20 k, 4 + 13 k, 2 + 6 k]] PISOT(20, 400 k + 134) = [[20, 400 k + 134], [7 + 20 k, -2 - 6 k]] 2 PISOT(20, 400 k + 135) = [[20, 400 k + 135, 8000 k + 5400 k + 911, 3 2 160000 k + 162000 k + 54665 k + 6148, 4 3 2 3200000 k + 4320000 k + 2186700 k + 491890 k + 41491], [8 + 20 k, -9 - 25 k, 4 + 11 k, -2 - 5 k, 2 + 6 k]] PISOT(20, 400 k + 145) = [ 2 [20, 400 k + 145, 8000 k + 5800 k + 1051], [8 + 20 k, -6 - 15 k, 4 + 11 k] ] 2 PISOT(20, 400 k + 146) = [[20, 400 k + 146, 8000 k + 5840 k + 1066, 3 2 160000 k + 175200 k + 63956 k + 7783], [7 + 20 k, 2 + 6 k, 2 + 4 k, -4 - 11 k]] 2 PISOT(20, 400 k + 149) = [[20, 400 k + 149, 8000 k + 5960 k + 1110, 3 2 160000 k + 178800 k + 66601 k + 8269], [7 + 20 k, 3 + 9 k, 3 + 7 k, -3 - 8 k]] 2 PISOT(20, 400 k + 150) = [[20, 400 k + 150, 8000 k + 6000 k + 1125, 3 2 160000 k + 180000 k + 67500 k + 8438], [8 + 20 k, -3 - 10 k, -6 - 15 k, 3 + 8 k]] PISOT(20, 400 k + 151) = [ 2 [20, 400 k + 151, 8000 k + 6040 k + 1140], [8 + 20 k, -3 - 9 k, -3 - 8 k]] 2 PISOT(20, 400 k + 153) = [[20, 400 k + 153, 8000 k + 6120 k + 1170], [9 + 20 k, -11 - 27 k, 5 + 13 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.995015191272054 2 PISOT(20, 400 k + 155) = [[20, 400 k + 155, 8000 k + 6200 k + 1201, 3 2 160000 k + 186000 k + 72065 k + 9306], [9 + 20 k, -11 - 25 k, 11 + 26 k, -7 - 18 k]] 2 PISOT(20, 400 k + 158) = [[20, 400 k + 158, 8000 k + 6320 k + 1248, 3 2 160000 k + 189600 k + 74884 k + 9858, 4 3 2 3200000 k + 5056000 k + 2995440 k + 788688 k + 77869], [8 + 20 k, -1 - 2 k, 1 + 4 k, 5 + 12 k, -2 - 5 k]] 2 PISOT(20, 400 k + 159) = [[20, 400 k + 159, 8000 k + 6360 k + 1264, 3 2 160000 k + 190800 k + 75841 k + 10048], [8 + 20 k, -1 - k, 5 + 12 k, -2 - 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.987649171158042 2 PISOT(20, 400 k + 161) = [[20, 400 k + 161, 8000 k + 6440 k + 1296, 3 2 160000 k + 193200 k + 77761 k + 10432, 4 3 2 5 3200000 k + 5152000 k + 3110460 k + 834592 k + 83971, 64000000 k 4 3 2 + 128800000 k + 103682400 k + 41730401 k + 8397560 k + 675913], [8 + 20 k, k, 3 + 8 k, 2 + 4 k, -3 - 8 k, -2 - 5 k]] 2 PISOT(20, 400 k + 162) = [[20, 400 k + 162, 8000 k + 6480 k + 1312, 3 2 160000 k + 194400 k + 78724 k + 10626, 4 3 2 3200000 k + 5184000 k + 3149040 k + 850128 k + 86061], [8 + 20 k, 1 + 2 k, -2 - 4 k, 3 + 8 k, 2 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.9746805881 PISOT(20, 400 k + 168) = [ 2 [20, 400 k + 168, 8000 k + 6720 k + 1411], [9 + 20 k, -6 - 12 k, 8 + 19 k] ] PISOT(20, 400 k + 171) = [ 2 [20, 400 k + 171, 8000 k + 6840 k + 1462], [8 + 20 k, 4 + 11 k, 6 + 14 k]] PISOT(20, 400 k + 175) = [ 2 [20, 400 k + 175, 8000 k + 7000 k + 1531], [9 + 20 k, -3 - 5 k, 7 + 16 k]] 2 PISOT(20, 400 k + 177) = [[20, 400 k + 177, 8000 k + 7080 k + 1566, 3 2 160000 k + 212400 k + 93969 k + 13855], [8 + 20 k, 8 + 17 k, -4 - 10 k, -4 - 9 k]] PISOT(20, 400 k + 178) = [[20, 400 k + 178], [8 + 20 k, 8 + 18 k]] 2 PISOT(20, 400 k + 179) = [[20, 400 k + 179, 8000 k + 7160 k + 1602, 3 2 160000 k + 214800 k + 96121 k + 14337], [8 + 20 k, 8 + 19 k, 4 + 10 k, 4 + 9 k]] PISOT(20, 400 k + 181) = 2 [[20, 400 k + 181, 8000 k + 7240 k + 1638], [9 + 20 k, 1 + k, -5 - 11 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.994067710148259 2 PISOT(20, 400 k + 184) = [[20, 400 k + 184, 8000 k + 7360 k + 1693, 3 2 160000 k + 220800 k + 101576 k + 15577, 4 3 2 5 3200000 k + 5888000 k + 4062960 k + 1246104 k + 143321, 64000000 k 4 3 2 + 147200000 k + 135430400 k + 62303344 k + 14331425 k + 1318669], [9 + 20 k, 1 + 4 k, 8 + 17 k, -2 - 4 k, 2 + 3 k, -6 - 13 k]] 2 PISOT(20, 400 k + 195) = [[20, 400 k + 195, 8000 k + 7800 k + 1901, 3 2 160000 k + 234000 k + 114065 k + 18532], [9 + 20 k, 7 + 15 k, 3 + 6 k, -1 - 2 k]] 2 PISOT(20, 400 k + 196) = [[20, 400 k + 196, 8000 k + 7840 k + 1921, 3 2 160000 k + 235200 k + 115256 k + 18828, 4 3 2 3200000 k + 6272000 k + 4610160 k + 1506152 k + 184536], [10 + 20 k, -2 - 4 k, 1 + k, -5 - 10 k, 1 + 2 k]] 2 PISOT(20, 400 k + 197) = [[20, 400 k + 197, 8000 k + 7880 k + 1940, 3 2 160000 k + 236400 k + 116409 k + 19105], [10 + 20 k, -1 - 3 k, -5 - 10 k, 1 + 2 k]] PISOT(20, 400 k + 198) = [[20, 400 k + 198], [10 + 20 k, -1 - 2 k]] PISOT(20, 400 k + 199) = 2 [[20, 400 k + 199, 8000 k + 7960 k + 1980], [10 + 20 k, -1 - k, 5 + 10 k]] PISOT(20, 400 k + 201) = 2 [[20, 400 k + 201, 8000 k + 8040 k + 2020], [10 + 20 k, k, 5 + 10 k]] PISOT(20, 400 k + 202) = [[20, 400 k + 202], [10 + 20 k, 1 + 2 k]] 2 PISOT(20, 400 k + 203) = [[20, 400 k + 203, 8000 k + 8120 k + 2060, 3 2 160000 k + 243600 k + 123609 k + 20904], [10 + 20 k, 2 + 3 k, -5 - 10 k, -1 - 2 k]] 2 PISOT(20, 400 k + 204) = [[20, 400 k + 204, 8000 k + 8160 k + 2081, 3 2 160000 k + 244800 k + 124856 k + 21228, 4 3 2 3200000 k + 6528000 k + 4994160 k + 1698168 k + 216544], [10 + 20 k, 2 + 4 k, k, 5 + 10 k, 1 + 2 k]] 2 PISOT(20, 400 k + 205) = [[20, 400 k + 205, 8000 k + 8200 k + 2101, 3 2 160000 k + 246000 k + 126065 k + 21533], [11 + 20 k, -8 - 15 k, 3 + 6 k, 1 + 2 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.994067548694371 2 PISOT(20, 400 k + 216) = [[20, 400 k + 216, 8000 k + 8640 k + 2333, 3 2 160000 k + 259200 k + 139976 k + 25199, 4 3 2 5 3200000 k + 6912000 k + 5598960 k + 2015816 k + 272177, 64000000 k 4 3 2 + 172800000 k + 186630400 k + 100787856 k + 27215937 k + 2939812], [11 + 20 k, -3 - 4 k, 9 + 17 k, 2 + 4 k, 1 + 3 k, 7 + 13 k]] PISOT(20, 400 k + 219) = 2 [[20, 400 k + 219, 8000 k + 8760 k + 2398], [11 + 20 k, -k, -6 - 11 k]] 2 PISOT(20, 400 k + 221) = [[20, 400 k + 221, 8000 k + 8840 k + 2442, 3 2 160000 k + 265200 k + 146521 k + 26984], [12 + 20 k, -11 - 19 k, 6 + 10 k, -5 - 9 k]] PISOT(20, 400 k + 222) = [[20, 400 k + 222], [12 + 20 k, -10 - 18 k]] 2 PISOT(20, 400 k + 223) = [[20, 400 k + 223, 8000 k + 8920 k + 2486, 3 2 160000 k + 267600 k + 149169 k + 27714], [12 + 20 k, -9 - 17 k, -6 - 10 k, 5 + 9 k]] PISOT(20, 400 k + 225) = [ 2 [20, 400 k + 225, 8000 k + 9000 k + 2531], [11 + 20 k, 2 + 5 k, 9 + 16 k]] 2 PISOT(20, 400 k + 229) = [[20, 400 k + 229, 8000 k + 9160 k + 2622], [12 + 20 k, -7 - 11 k, 8 + 14 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.9746781157 2 PISOT(20, 400 k + 232) = [[20, 400 k + 232, 8000 k + 9280 k + 2691], [11 + 20 k, 6 + 12 k, 11 + 19 k]] 2 PISOT(20, 400 k + 238) = [[20, 400 k + 238, 8000 k + 9520 k + 2832, 3 2 160000 k + 285600 k + 169924 k + 33698, 4 3 2 3200000 k + 7616000 k + 6797040 k + 2695952 k + 400973], [12 + 20 k, -1 - 2 k, -2 - 4 k, -5 - 8 k, 3 + 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.987649435817977 2 PISOT(20, 400 k + 239) = [[20, 400 k + 239, 8000 k + 9560 k + 2856, 3 2 160000 k + 286800 k + 171361 k + 34129, 4 3 2 5 3200000 k + 7648000 k + 6854460 k + 2730328 k + 407839, 64000000 k 4 3 2 + 191200000 k + 228482400 k + 136516799 k + 40783958 k + 4873646], [12 + 20 k, -1 - k, 5 + 8 k, -2 - 4 k, -5 - 8 k, 3 + 5 k]] 2 PISOT(20, 400 k + 241) = [[20, 400 k + 241, 8000 k + 9640 k + 2904, 3 2 160000 k + 289200 k + 174241 k + 34993], [12 + 20 k, k, 7 + 12 k, 3 + 5 k]] 2 PISOT(20, 400 k + 242) = [[20, 400 k + 242, 8000 k + 9680 k + 2928, 3 2 160000 k + 290400 k + 175684 k + 35426, 4 3 2 3200000 k + 7744000 k + 7027440 k + 2834192 k + 428621], [12 + 20 k, 1 + 2 k, 3 + 4 k, -7 - 12 k, -3 - 5 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.995014267642529 2 PISOT(20, 400 k + 245) = [[20, 400 k + 245, 8000 k + 9800 k + 3001, 3 2 160000 k + 294000 k + 180065 k + 36759], [11 + 20 k, 14 + 25 k, 15 + 26 k, 11 + 18 k]] 2 PISOT(20, 400 k + 247) = [[20, 400 k + 247, 8000 k + 9880 k + 3050], [11 + 20 k, 16 + 27 k, 8 + 13 k]] PISOT(20, 400 k + 249) = [ 2 [20, 400 k + 249, 8000 k + 9960 k + 3100], [12 + 20 k, 6 + 9 k, -5 - 8 k]] 2 PISOT(20, 400 k + 251) = [[20, 400 k + 251, 8000 k + 10040 k + 3150, 3 2 160000 k + 301200 k + 189001 k + 39532], [13 + 20 k, -6 - 9 k, 4 + 7 k, 5 + 8 k]] 2 PISOT(20, 400 k + 254) = [[20, 400 k + 254, 8000 k + 10160 k + 3226, 3 2 160000 k + 304800 k + 193556 k + 40973], [13 + 20 k, -4 - 6 k, 2 + 4 k, 7 + 11 k]] 2 PISOT(20, 400 k + 255) = [[20, 400 k + 255, 8000 k + 10200 k + 3251], [12 + 20 k, 9 + 15 k, 7 + 11 k]] 2 PISOT(20, 400 k + 265) = [[20, 400 k + 265, 8000 k + 10600 k + 3511, 3 2 160000 k + 318000 k + 210665 k + 46517, 4 3 2 3200000 k + 8480000 k + 8426700 k + 3721510 k + 616301], [12 + 20 k, 16 + 25 k, 7 + 11 k, 3 + 5 k, 4 + 6 k]] PISOT(20, 400 k + 266) = [[20, 400 k + 266], [13 + 20 k, 4 + 6 k]] 2 PISOT(20, 400 k + 267) = [[20, 400 k + 267, 8000 k + 10680 k + 3564], [14 + 20 k, -9 - 13 k, 4 + 6 k]] 2 PISOT(20, 400 k + 268) = [[20, 400 k + 268, 8000 k + 10720 k + 3591, 3 2 160000 k + 321600 k + 215464 k + 48117], [13 + 20 k, 5 + 8 k, 5 + 7 k, -4 - 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.999015314865883 2 PISOT(20, 400 k + 270) = [[20, 400 k + 270, 8000 k + 10800 k + 3645, 3 2 160000 k + 324000 k + 218700 k + 49208], [15 + 20 k, -21 - 30 k, 10 + 15 k, 2 + 3 k]] 2 PISOT(20, 400 k + 275) = [[20, 400 k + 275, 8000 k + 11000 k + 3781], [15 + 20 k, -18 - 25 k, 11 + 16 k]] 2 PISOT(20, 400 k + 281) = [[20, 400 k + 281, 8000 k + 11240 k + 3948, 3 2 160000 k + 337200 k + 236881 k + 55469], [13 + 20 k, 15 + 21 k, -3 - 5 k, -7 - 10 k]] PISOT(20, 400 k + 286) = [[20, 400 k + 286], [15 + 20 k, -10 - 14 k]] PISOT(20, 400 k + 287) = [[20, 400 k + 287], [14 + 20 k, 5 + 7 k]] 2 PISOT(20, 400 k + 290) = [[20, 400 k + 290, 8000 k + 11600 k + 4205, 3 2 160000 k + 348000 k + 252300 k + 60973, 4 3 2 3200000 k + 9280000 k + 10092000 k + 4877820 k + 884116], [15 + 20 k, -8 - 10 k, 11 + 15 k, -2 - 2 k, 8 + 11 k]] PISOT(20, 400 k + 291) = [[20, 400 k + 291], [14 + 20 k, 8 + 11 k]] 2 PISOT(20, 400 k + 298) = [[20, 400 k + 298, 8000 k + 11920 k + 4440, 3 2 160000 k + 357600 k + 266404 k + 66153, 4 3 2 3200000 k + 9536000 k + 10656240 k + 5292360 k + 985635], [15 + 20 k, -1 - 2 k, -8 - 10 k, 8 + 11 k, 3 + 4 k]] 2 PISOT(20, 400 k + 299) = [[20, 400 k + 299, 8000 k + 11960 k + 4470], [14 + 20 k, 14 + 19 k, 3 + 4 k]] 2 PISOT(20, 400 k + 302) = [[20, 400 k + 302, 8000 k + 12080 k + 4560, 3 2 160000 k + 362400 k + 273604 k + 68853, 4 3 2 3200000 k + 9664000 k + 10944240 k + 5508360 k + 1039635], [15 + 20 k, 1 + 2 k, 8 + 10 k, -7 - 9 k, 3 + 4 k]] PISOT(20, 400 k + 307) = [[20, 400 k + 307], [16 + 20 k, -10 - 13 k]] 2 PISOT(20, 400 k + 310) = [[20, 400 k + 310, 8000 k + 12400 k + 4805, 3 2 160000 k + 372000 k + 288300 k + 74478, 4 3 2 3200000 k + 9920000 k + 11532000 k + 5958220 k + 1154417], [16 + 20 k, -8 - 10 k, 4 + 5 k, -2 - 2 k, 7 + 9 k]] PISOT(20, 400 k + 311) = [[20, 400 k + 311], [16 + 20 k, -7 - 9 k]] 2 PISOT(20, 400 k + 312) = [[20, 400 k + 312, 8000 k + 12480 k + 4867, 3 2 160000 k + 374400 k + 292024 k + 75922], [15 + 20 k, 9 + 12 k, 5 + 7 k, 7 + 9 k]] PISOT(20, 400 k + 314) = [[20, 400 k + 314], [15 + 20 k, 11 + 14 k]] 2 PISOT(20, 400 k + 318) = [[20, 400 k + 318, 8000 k + 12720 k + 5056, 3 2 160000 k + 381600 k + 303364 k + 80387], [15 + 20 k, 14 + 18 k, 5 + 6 k, -4 - 5 k]] 2 PISOT(20, 400 k + 319) = [[20, 400 k + 319, 8000 k + 12760 k + 5088], [17 + 20 k, -17 - 21 k, 4 + 5 k]] 2 PISOT(20, 400 k + 322) = [[20, 400 k + 322, 8000 k + 12880 k + 5184, 3 2 160000 k + 386400 k + 311044 k + 83459], [15 + 20 k, 17 + 22 k, 11 + 14 k, 4 + 5 k]] 2 PISOT(20, 400 k + 323) = [[20, 400 k + 323, 8000 k + 12920 k + 5216, 3 2 160000 k + 387600 k + 312969 k + 84231, 4 3 2 3200000 k + 10336000 k + 12518940 k + 6738776 k + 1360211], [16 + 20 k, 2 + 3 k, 6 + 8 k, 7 + 9 k, 4 + 5 k]] 2 PISOT(20, 400 k + 325) = [[20, 400 k + 325, 8000 k + 13000 k + 5281], [17 + 20 k, -13 - 15 k, 13 + 16 k]] PISOT(20, 400 k + 328) = [ 2 [20, 400 k + 328, 8000 k + 13120 k + 5379], [16 + 20 k, 6 + 8 k, 9 + 11 k] ] 2 PISOT(20, 400 k + 332) = [[20, 400 k + 332, 8000 k + 13280 k + 5511, 3 2 160000 k + 398400 k + 330664 k + 91479, 4 3 2 3200000 k + 10624000 k + 13226640 k + 7318464 k + 1518492], [17 + 20 k, -7 - 8 k, 6 + 7 k, -3 - 4 k, -5 - 6 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.989005090299476 2 PISOT(20, 400 k + 333) = [[20, 400 k + 333, 8000 k + 13320 k + 5544, 3 2 160000 k + 399600 k + 332649 k + 92300, 4 3 2 3200000 k + 10656000 k + 13306140 k + 7384304 k + 1536668], [17 + 20 k, -6 - 7 k, 2 + 3 k, 8 + 10 k, 5 + 6 k]] PISOT(20, 400 k + 334) = [[20, 400 k + 334], [17 + 20 k, -5 - 6 k]] 2 PISOT(20, 400 k + 339) = [[20, 400 k + 339, 8000 k + 13560 k + 5746, 3 2 160000 k + 406800 k + 344761 k + 97394], [16 + 20 k, 15 + 19 k, 18 + 22 k, 11 + 13 k]] 2 PISOT(20, 400 k + 343) = [[20, 400 k + 343, 8000 k + 13720 k + 5882], [16 + 20 k, 19 + 23 k, 12 + 14 k]] 2 PISOT(20, 400 k + 349) = [[20, 400 k + 349, 8000 k + 13960 k + 6090], [18 + 20 k, -10 - 11 k, 7 + 8 k]] 2 PISOT(20, 400 k + 350) = [[20, 400 k + 350, 8000 k + 14000 k + 6125, 3 2 160000 k + 420000 k + 367500 k + 107188], [18 + 20 k, -9 - 10 k, 4 + 5 k, 7 + 8 k]] 2 PISOT(20, 400 k + 351) = [[20, 400 k + 351, 8000 k + 14040 k + 6160, 3 2 160000 k + 421200 k + 369601 k + 108107], [17 + 20 k, 9 + 11 k, 11 + 13 k, 7 + 8 k]] 2 PISOT(20, 400 k + 355) = [[20, 400 k + 355, 8000 k + 14200 k + 6301], [18 + 20 k, -4 - 5 k, -8 - 9 k]] PISOT(20, 400 k + 356) = [ 2 [20, 400 k + 356, 8000 k + 14240 k + 6337], [18 + 20 k, -4 - 4 k, 8 + 9 k] ] 2 PISOT(20, 400 k + 359) = [[20, 400 k + 359, 8000 k + 14360 k + 6444, 3 2 160000 k + 430800 k + 386641 k + 115669, 4 3 2 3200000 k + 11488000 k + 15465660 k + 9253552 k + 2076244], [18 + 20 k, -2 - k, 20 + 22 k, -5 - 5 k, 9 + 10 k]] 2 PISOT(20, 400 k + 363) = [[20, 400 k + 363, 8000 k + 14520 k + 6588], [19 + 20 k, -16 - 17 k, 10 + 11 k]] 2 PISOT(20, 400 k + 364) = [[20, 400 k + 364, 8000 k + 14560 k + 6625, 3 2 160000 k + 436800 k + 397496 k + 120579, 4 3 2 5 3200000 k + 11648000 k + 15899760 k + 9646160 k + 2194611, 64000000 k 4 3 2 + 291200000 k + 529990400 k + 482303344 k + 219456577 k + 39943252], [18 + 20 k, 3 + 4 k, 12 + 13 k, -3 - 3 k, 6 + 6 k, -10 - 11 k]] 2 PISOT(20, 400 k + 366) = [[20, 400 k + 366, 8000 k + 14640 k + 6698, 3 2 160000 k + 439200 k + 401876 k + 122577, 4 3 2 3200000 k + 11712000 k + 16074960 k + 9806016 k + 2243225], [19 + 20 k, -13 - 14 k, 4 + 4 k, -7 - 7 k, 11 + 12 k]] 2 PISOT(20, 400 k + 367) = [[20, 400 k + 367, 8000 k + 14680 k + 6734], [18 + 20 k, 7 + 7 k, -11 - 12 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.980902231713918 2 PISOT(20, 400 k + 372) = [[20, 400 k + 372, 8000 k + 14880 k + 6919, 3 2 160000 k + 446400 k + 415144 k + 128690, 4 3 2 3200000 k + 11904000 k + 16605840 k + 10295336 k + 2393571], [18 + 20 k, 11 + 12 k, 3 + 3 k, -3 - 4 k, -13 - 14 k]] Take it with a grain of salt, for k=1000 the Pisot index is, 0.983850698362542 2 PISOT(20, 400 k + 373) = [[20, 400 k + 373, 8000 k + 14920 k + 6956, 3 2 160000 k + 447600 k + 417369 k + 129721], [19 + 20 k, -6 - 7 k, -11 - 11 k, 14 + 15 k]] PISOT(20, 400 k + 377) = [[20, 400 k + 377], [18 + 20 k, 16 + 17 k]] PISOT(20, 400 k + 378) = [[20, 400 k + 378], [18 + 20 k, 17 + 18 k]] PISOT(20, 400 k + 379) = [[20, 400 k + 379], [18 + 20 k, 18 + 19 k]] 2 PISOT(20, 400 k + 388) = [[20, 400 k + 388, 8000 k + 15520 k + 7527, 3 2 160000 k + 465600 k + 451624 k + 146020, 4 3 2 5 3200000 k + 12416000 k + 18065040 k + 11681752 k + 2832714, 64000000 k 4 3 2 + 310400000 k + 602169600 k + 584092192 k + 283275809 k + 54953216], [20 + 20 k, -12 - 12 k, 7 + 7 k, -4 - 4 k, 2 + 2 k, -1 - k]] 2 PISOT(20, 400 k + 389) = [[20, 400 k + 389, 8000 k + 15560 k + 7566, 3 2 160000 k + 466800 k + 453961 k + 147158, 4 3 2 3200000 k + 12448000 k + 18158460 k + 11772668 k + 2862209], [20 + 20 k, -11 - 11 k, 6 + 6 k, -3 - 3 k, 1 + k]] 2 PISOT(20, 400 k + 390) = [[20, 400 k + 390, 8000 k + 15600 k + 7605, 3 2 160000 k + 468000 k + 456300 k + 148298, 4 3 2 3200000 k + 12480000 k + 18252000 k + 11863820 k + 2891821], [20 + 20 k, -10 - 10 k, 5 + 5 k, -2 - 2 k, 1 + k]] 2 PISOT(20, 400 k + 391) = [[20, 400 k + 391, 8000 k + 15640 k + 7644, 3 2 160000 k + 469200 k + 458641 k + 149439, 4 3 2 3200000 k + 12512000 k + 18345660 k + 11955168 k + 2921509], [20 + 20 k, -9 - 9 k, 4 + 4 k, -2 - 2 k, 1 + k]] 2 PISOT(20, 400 k + 392) = [[20, 400 k + 392, 8000 k + 15680 k + 7683, 3 2 160000 k + 470400 k + 460984 k + 150583], [20 + 20 k, -8 - 8 k, 3 + 3 k, -1 - k]] 2 PISOT(20, 400 k + 393) = [[20, 400 k + 393, 8000 k + 15720 k + 7722, 3 2 160000 k + 471600 k + 463329 k + 151728], [20 + 20 k, -7 - 7 k, 2 + 2 k, -1 - k]] 2 PISOT(20, 400 k + 394) = [[20, 400 k + 394, 8000 k + 15760 k + 7762, 3 2 160000 k + 472800 k + 465716 k + 152915], [20 + 20 k, -6 - 6 k, 2 + 2 k, -1 - k]] PISOT(20, 400 k + 395) = 2 [[20, 400 k + 395, 8000 k + 15800 k + 7801], [20 + 20 k, -5 - 5 k, 1 + k]] PISOT(20, 400 k + 396) = 2 [[20, 400 k + 396, 8000 k + 15840 k + 7841], [20 + 20 k, -4 - 4 k, 1 + k]] PISOT(20, 400 k + 397) = [[20, 400 k + 397], [20 + 20 k, -3 - 3 k]] PISOT(20, 400 k + 398) = [[20, 400 k + 398], [20 + 20 k, -2 - 2 k]] PISOT(20, 400 k + 399) = [[20, 400 k + 399], [20 + 20 k, -1 - k]] 2 The following pairs [a,b], PISOT(a, a k + b), do not posses recurrences of order<=, 6 {[4, 3], [4, 6], [4, 13], [5, 4], [5, 7], [5, 18], [5, 21], [6, 5], [6, 9], [6, 10], [6, 16], [6, 21], [6, 26], [6, 27], [6, 31], [7, 5], [7, 6], [7, 10], [7, 11], [7, 38], [7, 39], [7, 43], [7, 44], [8, 6], [8, 7], [8, 10], [8, 12], [8, 14], [8, 19], [8, 29], [8, 30], [8, 34], [8, 35], [8, 45], [8, 50], [8, 52], [8, 54], [8, 57], [8, 58], [9, 7], [9, 8], [9, 14], [9, 15], [9, 19], [9, 22], [9, 23], [9, 24], [9, 30], [9, 33], [9, 34], [9, 35], [9, 37], [9, 44], [9, 46], [9, 47], [9, 48], [9, 51], [9, 57], [9, 58], [9, 59], [9, 62], [9, 66], [9, 67], [9, 73], [9, 74], [10, 6], [10, 7], [10, 8], [10, 9], [10, 13], [10, 14], [10, 15], [10, 17], [10, 18], [10, 19], [10, 21], [10, 22], [10, 23], [10, 27], [10, 28], [10, 31], [10, 36], [10, 37], [10, 42], [10, 44], [10, 46], [10, 47], [10, 53], [10, 54], [10, 56], [10, 58], [10, 63], [10, 64], [10, 69], [10, 72], [10, 73], [10, 77], [10, 78], [10, 79], [10, 82], [10, 83], [10, 85], [10, 86], [10, 87], [10, 91], [10, 92], [10, 93], [11, 8], [11, 9], [11, 10], [11, 14], [11, 16], [11, 17], [11, 21], [11, 26], [11, 27], [11, 28], [11, 31], [11, 32], [11, 34], [11, 35], [11, 37], [11, 38], [11, 43], [11, 45], [11, 47], [11, 52], [11, 54], [11, 56], [11, 58], [11, 59], [11, 62], [11, 63], [11, 65], [11, 67], [11, 69], [11, 74], [11, 76], [11, 78], [11, 83], [11, 84], [11, 86], [11, 87], [11, 89], [11, 90], [11, 93], [11, 94], [11, 95], [11, 100], [11, 104], [11, 105], [11, 107], [11, 111], [11, 112], [11, 113], [12, 9], [12, 10], [12, 11], [12, 15], [12, 17], [12, 18], [12, 19], [12, 22], [12, 25], [12, 26], [12, 27], [12, 31], [12, 33], [12, 35], [12, 39], [12, 40], [12, 43], [12, 44], [12, 45], [12, 46], [12, 47], [12, 50], [12, 51], [12, 52], [12, 53], [12, 54], [12, 55], [12, 56], [12, 57], [12, 62], [12, 63], [12, 65], [12, 66], [12, 67], [12, 69], [12, 75], [12, 77], [12, 78], [12, 79], [12, 81], [12, 82], [12, 87], [12, 88], [12, 89], [12, 91], [12, 92], [12, 93], [12, 94], [12, 97], [12, 98], [12, 99], [12, 100], [12, 101], [12, 104], [12, 105], [12, 109], [12, 111], [12, 113], [12, 117], [12, 118], [12, 119], [12, 122], [12, 125], [12, 127], [12, 129], [12, 133], [12, 134], [12, 135], [13, 9], [13, 10], [13, 11], [13, 12], [13, 16], [13, 17], [13, 18], [13, 20], [13, 25], [13, 27], [13, 30], [13, 31], [13, 36], [13, 40], [13, 42], [13, 45], [13, 47], [13, 49], [13, 50], [13, 53], [13, 54], [13, 60], [13, 62], [13, 63], [13, 67], [13, 70], [13, 73], [13, 74], [13, 76], [13, 77], [13, 79], [13, 80], [13, 81], [13, 88], [13, 89], [13, 90], [13, 92], [13, 93], [13, 95], [13, 96], [13, 99], [13, 102], [13, 106], [13, 107], [13, 109], [13, 115], [13, 116], [13, 119], [13, 120], [13, 122], [13, 124], [13, 127], [13, 129], [13, 133], [13, 138], [13, 139], [13, 142], [13, 144], [13, 149], [13, 151], [13, 152], [13, 153], [13, 157], [13, 158], [13, 159], [13, 160], [14, 9], [14, 10], [14, 11], [14, 12], [14, 13], [14, 17], [14, 18], [14, 20], [14, 21], [14, 22], [14, 23], [14, 24], [14, 25], [14, 26], [14, 27], [14, 30], [14, 32], [14, 34], [14, 36], [14, 37], [14, 41], [14, 46], [14, 52], [14, 54], [14, 55], [14, 57], [14, 58], [14, 61], [14, 62], [14, 64], [14, 69], [14, 72], [14, 73], [14, 74], [14, 76], [14, 80], [14, 81], [14, 86], [14, 89], [14, 90], [14, 91], [14, 92], [14, 93], [14, 94], [14, 95], [14, 101], [14, 102], [14, 103], [14, 104], [14, 105], [14, 106], [14, 107], [14, 110], [14, 115], [14, 116], [14, 120], [14, 122], [14, 123], [14, 124], [14, 127], [14, 132], [14, 134], [14, 135], [14, 138], [14, 139], [14, 141], [14, 142], [14, 144], [14, 150], [14, 155], [14, 159], [14, 160], [14, 162], [14, 164], [14, 166], [14, 169], [14, 170], [14, 171], [14, 172], [14, 173], [14, 174], [14, 175], [14, 176], [14, 178], [14, 179], [14, 183], [14, 184], [14, 185], [14, 186], [14, 187], [15, 10], [15, 11], [15, 12], [15, 13], [15, 14], [15, 18], [15, 19], [15, 23], [15, 24], [15, 25], [15, 26], [15, 27], [15, 34], [15, 35], [15, 36], [15, 37], [15, 39], [15, 40], [15, 42], [15, 43], [15, 46], [15, 47], [15, 48], [15, 49], [15, 50], [15, 51], [15, 52], [15, 53], [15, 59], [15, 62], [15, 63], [15, 66], [15, 67], [15, 68], [15, 69], [15, 70], [15, 71], [15, 77], [15, 79], [15, 80], [15, 81], [15, 83], [15, 86], [15, 88], [15, 89], [15, 92], [15, 93], [15, 94], [15, 100], [15, 101], [15, 103], [15, 104], [15, 106], [15, 107], [15, 108], [15, 110], [15, 111], [15, 114], [15, 115], [15, 117], [15, 118], [15, 119], [15, 121], [15, 122], [15, 124], [15, 125], [15, 131], [15, 132], [15, 133], [15, 136], [15, 137], [15, 139], [15, 142], [15, 144], [15, 145], [15, 146], [15, 148], [15, 154], [15, 155], [15, 156], [15, 157], [15, 158], [15, 159], [15, 162], [15, 163], [15, 166], [15, 171], [15, 172], [15, 173], [15, 174], [15, 175], [15, 176], [15, 177], [15, 178], [15, 179], [15, 182], [15, 183], [15, 185], [15, 186], [15, 188], [15, 189], [15, 190], [15, 191], [15, 198], [15, 199], [15, 200], [15, 201], [15, 202], [15, 206], [15, 207], [15, 211], [15, 212], [15, 213], [15, 214], [15, 215], [16, 10], [16, 11], [16, 12], 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This ends this fascinating article that tooks, 2900.303, to generated.