###################################################################### ##CatalanCC: Save this file as CatalanCC # ## To use it, stay in the # ##same directory, get into Maple (by typing: maple ) # ##and then type: read CatalanCC # ##Then follow the instructions given there # ## # ##Written by Doron Zeilberger, Rutgers University , # #zeilberg at math dot rutgers dot edu # ###################################################################### #Created: Oct. 2013 print(`Created: Oct. 2013`): print(` This is CatalanCC `): print(`To compute FAST (in log. time in n) the Catalan numbers modulo m for ALL m divisible by `, ifactor(2768774904222066200260800) ): print(`It is one of the packages that accompany the article `): print(`A Case Study in Meta-Automation: Automatic Generation of Automata For Proving Congruence Properties of Combinatorial Sequences `): print(`by Eric Rowland and Doron Zeilberger`): print(`available from Rowland's and Zeilberger's websites and arxiv.org`): print(``): print(`Please report bugs to zeilberg at math dot rutgers dot edu`): print(``): print(`The most current version of this package and paper`): print(` are available from`): print(`http://www.math.rutgers.edu/~zeilberg/ .`): with(combinat): with(numtheory): ezra:=proc() if args=NULL then print(`The main procedures are: `): print(` CatModM, CatModMseq `): print(` `): elif nops([args])=1 and op(1,[args])=CatModM then print(` CatModM(n,m): inputs integers n and m, and returns the n-th Catalan number mod m `): print(`m must be a divisor of `, ifactor(2768774904222066200260800)): print(`For example, to find the last two decimal digits of the 100000's Catalan number, type:`): print(`Try: CatModM(100000,100); `): elif nops([args])=1 and op(1,[args])=CatModMseq then print(` CatModMseq(N,m): inputs integers N and m, and returns the first N terms of the sequence n-th Catalan number mod m `): print(`m must be a divisor of `, ifactor(2768774904222066200260800)): print(`For example, to find the sequence of the first 1000 Catalan numbers mod 100, type:`): print(`Try: CatModMseq(1000,100); `): else print(`There is no ezra for`,args): fi: end: ######### #EvalEtzim(L,n1,m) : inputs a list of lists of congruence trees,L, such that L[i][j] gives #the scheme for mod ithprime(i)^j, finds it mod m, if possible #Try: L:=Etzim([x+2+1/x,1-x],x,100); #EvalEtzim(L,100,12); EvalEtzim:=proc(L1,n1,m) local i, M,ku,j,p,lu1,lu2,L: if not type(L1[2]/m, integer) then RETURN(FAIL): fi: L:=L1[1]: ku:=[]: for i from 1 to nops(L) do p:=ithprime(i): if type(m/p, integer) then for j from 1 while type(m/p^j,integer) do od: ku:=[op(ku),[i,j-1]]: fi: od: lu1:=[]: lu2:=[]: for i from 1 to nops(ku) do lu2:=[op(lu2),ithprime(ku[i][1])^ku[i][2]]: lu1:=[op(lu1),EvalEtzp(L[ku[i][1]][ku[i][2]],n1,1,ithprime(ku[i][1]))]: od: chrem(lu1,lu2): end: #SeqEtzim(L,N,m): the first N terms of the mod m #of the sequence generated by the multi-schem L #Try: #L:=Etzim([x+2+1/x,1-x],x,100); #EvalEtzim(L,100,12); SeqEtzim:=proc(L,N,m) local n1: if not type(L[2]/m, integer) then print(`can't handle modulo`, m): RETURN(FAIL): fi: [seq(EvalEtzim(L,n1,m) , n1=1..N)]: end: ##### CatModM:=proc(n1,m) EvalEtzim(Lcatalan,n1,m) : end: CatModMseq:=proc(N,m) SeqEtzim(Lcatalan,N,m): end: #EvalEtzp(T,n1,i,p): inputs a p-ary congruence tree T, a positive intger n1, and an integer i between 1 and nops(T[1]) #and outputs the value of the i-th vertex at n1. Try: #EvalEtzp([[[2, 1], [2, 3], [3, 0]], [1, 2, 0]],10,1,2); EvalEtzp:=proc(T,n1,i,p) local ka: option remember: if n1=0 then RETURN(T[2][i]): fi: ka:=n1 mod p: if T[1][i][ka+1]=0 then RETURN(0): else RETURN(EvalEtzp(T,(n1-ka)/p,T[1][i][ka+1] , p) ) : fi: end: Lcatalan:= [[[[[[2, 1], [2, 0]], [1, 1]], [[[2, 1], [2, 3], [3, 0]], [1, 1, 2]], [[[2, 3], [4, 5], [2, 6], [4, 7], [8, 9], [10, 11], [12, 13], [8, 13], [15, 0], [16, 17], [10, 11], [12, 9], [15, 0], [15, 0], [15, 0], [16, 20], [8, 9], [15, 0], [15, 0 ], [12, 13]], [1, 1, 1, 1, 2, 5, 6, 2, 4, 5, 5, 6, 4, 4, 4, 5, 2, 4, 4, 6]], [[ [2, 3], [4, 5], [6, 7], [8, 9], [10, 11], [4, 12], [13, 14], [8, 15], [16, 17], [18, 19], [20, 21], [22, 23], [24, 25], [26, 27], [28, 29], [30, 31], [32, 33], [18, 34], [32, 36], [37, 38], [39, 40], [41, 34], [43, 44], [45, 46], [47, 48], [49, 50], [51, 52], [53, 54], [55, 36], [30, 54], [20, 44], [59, 60], [61, 62], [55, 33], [59, 60], [61, 62], [37, 38], [61, 62], [71, 72], [0, 0], [41, 19], [ 55, 33], [37, 38], [39, 40], [45, 74], [75, 76], [18, 19], [20, 21], [77, 78], [79, 80], [49, 50], [51, 52], [53, 31], [43, 21], [59, 60], [61, 62], [43, 21], [39, 40], [59, 60], [39, 40], [71, 72], [0, 0], [59, 60], [59, 60], [39, 40], [ 71, 72], [0, 0], [61, 62], [71, 72], [0, 0], [71, 72], [0, 0], [59, 60], [93, 94], [53, 54], [55, 36], [77, 95], [96, 97], [18, 19], [20, 21], [59, 60], [39, 40], [71, 72], [0, 0], [39, 40], [71, 72], [0, 0], [71, 72], [0, 0], [71, 72], [0, 0], [0, 0], [30, 31], [32, 33], [98, 99], [53, 54], [55, 36], [30, 31], [32 , 33]], [1, 1, 1, 1, 2, 1, 5, 1, 14, 2, 4, 10, 5, 13, 6, 14, 12, 2, 12, 4, 8, 10, 4, 5, 2, 13, 13, 6, 12, 14, 4, 12, 8, 12, 12, 8, 4, 8, 8, 0, 10, 12, 4, 8, 5, 6, 2, 4, 13, 2, 13, 13, 6, 4, 12, 8, 4, 8, 12, 8, 8, 0, 12, 12, 8, 8, 0, 8, 8, 0, 8, 0, 12, 14, 6, 12, 13, 6, 2, 4, 12, 8, 8, 0, 8, 8, 0, 8, 0, 8, 0, 0, 14 , 12, 14, 6, 12, 14, 12]], [[[2, 3], [4, 5], [6, 7], [8, 9], [10, 11], [12, 13] , [14, 15], [16, 17], [18, 19], [20, 21], [22, 23], [8, 24], [25, 26], [27, 28] , [29, 30], [16, 31], [32, 33], [34, 35], [36, 37], [38, 39], [40, 41], [42, 43 ], [44, 45], [46, 47], [48, 49], [50, 51], [52, 53], [54, 55], [56, 57], [58, 59], [60, 61], [62, 63], [64, 65], [66, 67], [68, 69], [70, 71], [72, 73], [38, 74], [75, 76], [77, 78], [79, 80], [81, 82], [83, 84], [85, 86], [87, 88], [89, 90], [91, 92], [93, 94], [75, 96], [97, 98], [99, 100], [101, 102], [103, 104], [105, 106], [107, 108], [109, 110], [111, 112], [113, 114], [115, 116], [117, 118], [119, 120], [121, 122], [123, 124], [125, 126], [127, 128], [66, 129], [ 130, 124], [132, 133], [134, 135], [136, 137], [138, 139], [140, 141], [142, 143], [144, 96], [146, 147], [148, 149], [150, 151], [152, 153], [140, 141], [ 142, 143], [81, 82], [159, 160], [140, 141], [163, 164], [165, 166], [167, 168] , [169, 170], [171, 172], [173, 129], [175, 176], [177, 178], [179, 180], [93, 181], [182, 41], [146, 147], [185, 186], [187, 188], [159, 160], [191, 192], [ 193, 194], [101, 195], [196, 197], [198, 199], [200, 201], [38, 39], [40, 41], [42, 43], [44, 45], [202, 203], [204, 205], [206, 207], [208, 209], [210, 211], [212, 213], [214, 215], [216, 217], [218, 219], [130, 221], [222, 223], [224, 225], [121, 219], [68, 176], [228, 229], [230, 231], [150, 151], [152, 153], [ 236, 237], [163, 164], [123, 221], [241, 242], [230, 231], [81, 82], [83, 84], [191, 192], [193, 194], [136, 137], [152, 153], [85, 86], [193, 194], [254, 255 ], [256, 257], [258, 259], [260, 261], [262, 263], [185, 186], [136, 137], [138 , 139], [236, 237], [163, 164], [150, 151], [138, 139], [191, 192], [87, 88], [ 254, 255], [256, 257], [258, 259], [260, 261], [159, 160], [236, 237], [142, 143], [254, 255], [256, 257], [258, 259], [260, 261], [165, 166], [167, 168], [ 258, 259], [260, 261], [300, 301], [302, 303], [0, 0], [0, 0], [173, 67], [123, 221], [305, 306], [307, 308], [136, 137], [138, 139], [140, 141], [142, 143], [ 40, 309], [310, 311], [136, 137], [138, 139], [236, 237], [163, 164], [187, 188 ], [83, 84], [236, 237], [142, 143], [165, 166], [167, 168], [169, 170], [171, 172], [317, 318], [319, 320], [321, 322], [121, 122], [123, 124], [125, 126], [ 127, 128], [202, 323], [324, 325], [326, 327], [328, 329], [330, 74], [182, 309 ], [332, 333], [334, 335], [336, 337], [338, 339], [340, 341], [342, 343], [210 , 211], [212, 213], [214, 215], [216, 217], [218, 122], [175, 69], [241, 242], [346, 347], [150, 151], [152, 153], [236, 237], [163, 164], [175, 69], [307, 308], [187, 188], [159, 160], [85, 86], [87, 88], [150, 151], [138, 139], [191, 192], [87, 88], [254, 255], [256, 257], [258, 259], [260, 261], [346, 347], [ 187, 188], [159, 160], [85, 86], [87, 88], [165, 166], [167, 168], [169, 170], [171, 172], [152, 153], [191, 192], [87, 88], [169, 170], [171, 172], [254, 255 ], [256, 257], [169, 170], [171, 172], [300, 301], [302, 303], [0, 0], [0, 0], [136, 137], [138, 139], [236, 237], [163, 164], [254, 255], [256, 257], [258, 259], [260, 261], [138, 139], [165, 166], [167, 168], [254, 255], [256, 257], [ 169, 170], [171, 172], [300, 301], [302, 303], [0, 0], [0, 0], [236, 237], [142 , 143], [254, 255], [256, 257], [254, 255], [256, 257], [169, 170], [171, 172], [300, 301], [302, 303], [0, 0], [0, 0], [167, 168], [258, 259], [260, 261], [ 300, 301], [302, 303], [0, 0], [0, 0], [300, 301], [302, 303], [0, 0], [0, 0], [346, 347], [81, 82], [83, 84], [191, 192], [193, 194], [386, 387], [150, 151], [152, 153], [254, 255], [256, 257], [83, 84], [254, 255], [256, 257], [388, 389 ], [390, 391], [66, 67], [68, 69], [70, 71], [72, 73], [392, 393], [394, 395], [396, 397], [218, 219], [130, 221], [222, 223], [224, 225], [330, 39], [386, 387], [81, 82], [83, 84], [85, 86], [87, 88], [336, 400], [401, 402], [403, 404 ], [405, 406], [330, 74], [182, 309], [332, 333], [334, 335], [187, 188], [159, 160], [85, 86], [87, 88], [254, 255], [256, 257], [191, 192], [193, 194], [254, 255], [256, 257], [169, 170], [171, 172], [300, 301], [302, 303], [0, 0], [0, 0 ], [85, 86], [87, 88], [256, 257], [169, 170], [171, 172], [300, 301], [302, 303], [0, 0], [0, 0], [300, 301], [302, 303], [0, 0], [0, 0], [300, 301], [302, 303], [0, 0], [0, 0], [300, 301], [302, 303], [0, 0], [0, 0], [302, 303], [0, 0 ], [0, 0], [85, 86], [87, 88], [140, 141], [142, 143], [173, 129], [175, 176], [177, 178], [179, 180], [407, 408], [409, 410], [173, 129], [175, 176], [177, 178], [179, 180], [140, 141], [142, 143], [411, 412], [413, 414], [415, 416], [ 218, 219], [130, 221], [222, 223], [224, 225], [66, 67], [68, 69], [70, 71], [ 72, 73], [417, 418], [419, 420], [173, 129], [175, 176], [177, 178], [179, 180] , [66, 67], [68, 69], [70, 71], [72, 73]], [1, 1, 1, 1, 2, 1, 5, 1, 14, 2, 4, 1 , 10, 5, 13, 1, 22, 14, 12, 2, 28, 4, 8, 30, 10, 20, 5, 2, 13, 29, 6, 22, 28, 14, 4, 12, 24, 2, 12, 28, 24, 4, 24, 8, 16, 30, 12, 10, 12, 20, 8, 5, 22, 2, 4, 13, 18, 29, 29, 6, 28, 22, 20, 28, 24, 14, 20, 4, 8, 12, 8, 24, 16, 12, 12, 24, 28, 8, 24, 16, 4, 24, 24, 16, 8, 16, 16, 0, 30, 4, 12, 24, 10, 28, 12, 24, 20, 24, 8, 16, 5, 14, 22, 28, 2, 28, 4, 8, 13, 6, 18, 4, 29, 18, 29, 29, 6, 20, 28, 24, 22, 4, 20, 8, 28, 8, 24, 16, 20, 20, 8, 4, 24, 8, 16, 12, 8, 8, 16, 24, 16, 16, 0, 12, 24, 12, 8, 24, 16, 28, 8, 8, 16, 24, 16, 16, 0, 24, 24, 16, 24, 16, 16, 0, 8, 16, 16, 0, 16, 0, 0, 0, 30, 20, 4, 8, 12, 8, 24, 16, 28, 28, 12, 8, 24, 16, 20, 24, 24, 16, 8, 16, 16, 0, 30, 14, 12, 22, 20, 28, 24, 13, 30, 6, 28 , 18, 28, 4, 8, 29, 6, 18, 4, 29, 18, 29, 29, 6, 4, 20, 8, 28, 8, 24, 16, 4, 8, 20, 24, 8, 16, 28, 8, 8, 16, 24, 16, 16, 0, 8, 20, 24, 8, 16, 8, 16, 16, 0, 8, 8, 16, 16, 0, 24, 16, 16, 0, 16, 0, 0, 0, 12, 8, 24, 16, 24, 16, 16, 0, 8, 8, 16, 24, 16, 16, 0, 16, 0, 0, 0, 24, 16, 24, 16, 24, 16, 16, 0, 16, 0, 0, 0, 16, 16, 0, 16, 0, 0, 0, 16, 0, 0, 0, 8, 4, 24, 8, 16, 24, 28, 8, 24, 16, 24, 24, 16 , 30, 12, 14, 4, 12, 24, 14, 30, 12, 6, 20, 28, 24, 18, 24, 4, 24, 8, 16, 29, 30, 6, 28, 18, 28, 4, 8, 20, 24, 8, 16, 24, 16, 8, 16, 24, 16, 16, 0, 16, 0, 0, 0, 8, 16, 16, 16, 0, 16, 0, 0, 0, 16, 0, 0, 0, 16, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 8, 16, 24, 16, 30, 4, 12, 24, 14, 12, 30, 4, 12, 24, 24, 16, 14, 30, 12, 6, 20, 28, 24, 14, 4, 12, 24, 14, 12, 30, 4, 12, 24, 14, 4, 12, 24]], [[[2, 3], [4 , 5], [6, 7], [8, 9], [10, 11], [12, 13], [14, 15], [16, 17], [18, 19], [20, 21 ], [22, 23], [24, 25], [26, 27], [28, 29], [30, 31], [32, 33], [34, 35], [36, 37], [38, 39], [40, 41], [42, 43], [44, 45], [46, 47], [16, 48], [49, 50], [51, 52], [53, 54], [55, 56], [57, 58], [59, 60], [61, 62], [32, 63], [64, 65], [66, 67], [68, 69], [70, 71], [72, 73], [74, 75], [76, 77], [78, 79], [80, 81], [82, 83], [84, 85], [86, 87], [88, 89], [90, 91], [92, 93], [94, 95], [96, 97], [98, 99], [100, 101], [102, 103], [104, 105], [106, 107], [108, 109], [110, 111], [ 112, 113], [114, 115], [116, 117], [118, 119], [120, 121], [122, 123], [124, 125], [126, 127], [128, 129], [130, 131], [132, 133], [134, 135], [136, 137], [ 138, 139], [140, 141], [142, 143], [144, 145], [146, 147], [148, 149], [150, 151], [152, 153], [78, 154], [155, 156], [157, 158], [159, 160], [161, 162], [ 163, 164], [165, 166], [167, 168], [169, 170], [171, 172], [173, 174], [175, 176], [177, 178], [179, 180], [181, 182], [183, 184], [185, 186], [187, 188], [ 189, 190], [191, 192], [193, 194], [195, 196], [197, 198], [199, 200], [201, 202], [203, 204], [205, 206], [207, 208], [209, 210], [211, 212], [213, 214], [ 215, 216], [217, 218], [219, 220], [221, 222], [223, 224], [225, 226], [227, 228], [229, 230], [231, 232], [233, 234], [235, 236], [237, 238], [239, 240], [ 241, 242], [243, 244], [245, 246], [247, 248], [249, 250], [251, 252], [253, 254], [255, 256], [257, 258], [259, 260], [261, 262], [263, 264], [265, 266], [ 267, 268], [269, 270], [271, 272], [138, 273], [274, 275], [276, 277], [278, 279], [280, 281], [282, 283], [284, 285], [286, 287], [288, 289], [290, 291], [ 292, 293], [294, 295], [296, 297], [298, 299], [300, 301], [302, 303], [304, 305], [306, 307], [308, 309], [310, 311], [312, 313], [314, 315], [316, 317], [ 318, 319], [320, 321], [322, 323], [324, 325], [326, 327], [328, 329], [300, 301], [302, 303], [169, 170], [335, 336], [337, 338], [339, 340], [341, 342], [ 343, 344], [345, 346], [347, 348], [349, 350], [351, 352], [353, 354], [355, 356], [357, 358], [359, 360], [361, 362], [363, 364], [365, 366], [367, 368], [ 369, 370], [371, 372], [373, 374], [274, 376], [377, 378], [379, 380], [381, 382], [383, 384], [385, 386], [387, 388], [197, 389], [390, 391], [392, 393], [ 394, 395], [396, 397], [398, 399], [400, 401], [402, 403], [404, 405], [406, 407], [337, 338], [410, 411], [412, 413], [414, 415], [416, 417], [418, 419], [ 213, 420], [421, 422], [423, 424], [425, 426], [427, 428], [429, 430], [431, 432], [433, 434], [435, 154], [436, 437], [438, 439], [440, 441], [442, 443], [ 444, 445], [446, 447], [448, 449], [450, 451], [452, 453], [454, 455], [456, 457], [458, 459], [460, 461], [462, 463], [464, 465], [466, 467], [468, 469], [ 470, 471], [472, 473], [474, 475], [476, 477], [478, 479], [480, 481], [482, 483], [484, 485], [486, 487], [488, 489], [490, 491], [492, 493], [494, 495], [ 496, 497], [498, 499], [500, 501], [502, 503], [504, 505], [257, 506], [507, 493], [509, 510], [511, 512], [513, 514], [515, 516], [517, 518], [519, 520], [ 521, 522], [523, 524], [322, 323], [324, 325], [529, 530], [531, 532], [533, 534], [535, 536], [537, 376], [539, 540], [541, 542], [543, 544], [545, 546], [ 547, 548], [519, 520], [551, 552], [335, 336], [173, 174], [175, 176], [555, 556], [557, 558], [416, 417], [418, 419], [288, 289], [564, 565], [566, 567], [ 324, 325], [570, 571], [572, 573], [574, 575], [576, 577], [578, 579], [580, 581], [582, 583], [584, 585], [586, 587], [588, 589], [590, 591], [592, 593], [ 594, 595], [596, 597], [598, 599], [600, 601], [602, 603], [604, 605], [606, 607], [564, 565], [292, 293], [294, 295], [610, 611], [612, 613], [533, 534], [ 535, 536], [318, 319], [523, 524], [621, 622], [294, 295], [623, 624], [625, 626], [627, 628], [629, 630], [578, 579], [580, 581], [582, 583], [584, 585], [ 586, 587], [588, 589], [590, 591], [592, 593], [335, 336], [649, 650], [410, 411], [653, 654], [655, 656], [657, 658], [659, 660], [661, 662], [663, 664], [ 665, 666], [667, 668], [586, 587], [588, 589], [590, 591], [592, 593], [349, 350], [351, 352], [680, 681], [682, 683], [586, 587], [588, 589], [688, 689], [ 690, 691], [692, 693], [694, 695], [696, 697], [698, 699], [700, 701], [702, 703], [704, 705], [706, 707], [708, 506], [710, 711], [712, 713], [714, 715], [ 716, 717], [718, 719], [720, 721], [722, 723], [373, 724], [725, 141], [539, 540], [728, 729], [730, 731], [732, 733], [734, 735], [736, 737], [606, 607], [ 564, 565], [621, 622], [744, 745], [326, 327], [328, 329], [746, 747], [748, 749], [750, 751], [752, 753], [754, 755], [756, 757], [758, 759], [760, 761], [ 762, 763], [288, 289], [290, 291], [621, 622], [744, 745], [610, 611], [612, 613], [533, 534], [535, 536], [404, 405], [777, 778], [779, 780], [175, 176], [ 653, 654], [655, 656], [785, 786], [787, 788], [789, 790], [791, 792], [680, 681], [682, 683], [797, 798], [799, 800], [801, 802], [803, 804], [805, 806], [ 807, 808], [809, 810], [811, 812], [813, 814], [815, 816], [817, 818], [257, 258], [259, 260], [261, 262], [263, 264], [819, 820], [821, 822], [823, 824], [ 825, 826], [435, 79], [827, 828], [829, 830], [831, 832], [833, 834], [835, 836 ], [837, 838], [551, 552], [335, 336], [779, 780], [843, 844], [845, 846], [847 , 848], [849, 850], [851, 852], [450, 853], [854, 855], [856, 857], [858, 859], [860, 861], [862, 863], [864, 865], [866, 867], [868, 869], [155, 305], [870, 871], [872, 873], [874, 875], [876, 877], [878, 879], [880, 881], [882, 883], [ 884, 885], [886, 887], [888, 889], [890, 891], [892, 893], [894, 895], [896, 897], [898, 899], [900, 901], [902, 903], [904, 905], [906, 907], [908, 909], [ 910, 911], [912, 913], [914, 915], [507, 917], [918, 919], [920, 921], [922, 923], [924, 925], [926, 927], 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [104, 20, 242, 23, 15, 16, 40, 119, 102, 13, 2, 138, 35, 42, 27, 49, 7, 25, 60, 25, 6, 104, 23, 35, 38, 9, 15, 19, 27, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [102, 16, 43, 98, 29, 119, 5, 42, 31, 60, 40, 25, 7, 38, 11, 13, 58, 27, 23, 27, 150, 102, 98, 7, 125 , 39, 29, 227, 11, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [150, 49, 21, 31, 25, 13, 4, 60, 20, 104, 30, 44 , 5, 23, 127, 99, 12, 48, 102, 48, 242, 150, 31, 5, 98, 41, 25, 172, 127, 24, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [107, 14, 25, 9, 99, 2, 167, 40, 47, 30, 31, 128, 19, 5, 150, 11, 227, 6 , 4, 6, 15, 107, 9, 19, 12, 42, 99, 45, 150, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [125, 227, 167, 127, 39, 138, 6, 25, 22, 44, 128, 41, 104, 27, 35, 111, 102, 18, 48, 18, 98, 125, 127, 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [150, 49, 21, 31, 25, 13, 4, 60, 20, 104, 30, 44, 5, 23, 127, 99, 12, 48, 102 , 48, 242, 150, 31, 5, 98, 41, 25, 172, 127, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [157, 6, 7, 13, 22, 150, 14, 20, 99, 43, 29, 19, 41, 16, 138, 242, 18, 227, 49, 227, 55, 157, 13, 41, 119, 30, 22, 23, 138, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [98, 172, 16, 22, 41, 111, 58, 44, 19, 38, 7, 14, 6, 48, 40, 42, 150, 2, 125, 2, 31, 98, 22, 6, 127, 21, 41, 15, 40, 5 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [242, 157, 45, 20, 44, 99, 107, 104, 43, 6, 24, 38, 4, 102, 22, 128, 9, 125, 150, 125, 144, 242, 20, 4, 31, 14, 44, 119, 22, 103, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [144, 55, 18, 43, 38, 128, 15, 6, 21, 58, 103, 23, 107, 150, 19, 7, 47, 98, 242, 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