###################################################################### ##MotzkinCC: Save this file as MotzkinCC # ## To use it, stay in the # ##same directory, get into Maple (by typing: maple ) # ##and then type: read MotzkinCC # ##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 MotzkinCC `): print(`To compute FAST (in log. time in n) the Motzkin numbers modulo m for ALL m divisible by `, ifactor(232908956280) ): 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(` MotzkinModM, MotzkinModMseq `): print(` `): elif nops([args])=1 and op(1,[args])=MotzkinModM then print(` MotzkinModM(n,m): inputs integers n and m, and returns the n-th Motzkin number mod m `): print(`m must be a divisor of `, ifactor(232908956280)): print(`For example, to find the 100000's Motzkin number mod 20, type:`): print(`Try: MotzkinModM(100000,20); `): elif nops([args])=1 and op(1,[args])=MotzkinModMseq then print(` MotzkinModMseq(N,m): inputs integers N and m, and returns the first N terms of the sequence n-th Motzkin number mod m `): print(`m must be a divisor of `, ifactor(232908956280)): print(`For example, to find the sequence of the first 1000 Motzkin numbers mod 20, type:`): print(`Try: MotzkinModMseq(1000,20); `): 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: #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: MotzkinModM:=proc(n1,m) EvalEtzim(Lmotzkin,n1,m) : end: MotzkinModMseq:=proc(N,m) SeqEtzim(Lmotzkin,N,m): end: Lmotzkin:= [[[[[[2, 2], [4, 5], [4, 5], [4, 4], [0, 2], [4, 4], [0, 2]], [1, 1, 1, 1, 0, 1 , 0]], [[[2, 3], [4, 5], [4, 7], [8, 9], [10, 11], [8, 9], [14, 15], [8, 16], [ 8, 17], [18, 0], [19, 20], [8, 16], [8, 17], [0, 23], [24, 7], [25, 26], [25, 26], [18, 29], [25, 31], [10, 11], [25, 26], [25, 26], [18, 29], [25, 31], [25, 39], [8, 17], [25, 39], [8, 17], [18, 29], [25, 39], [25, 26], [25, 39], [8, 17 ], [25, 39], [8, 17], [18, 29], [25, 39], [25, 26], [8, 17], [8, 17], [8, 17], [8, 17], [8, 17], [8, 17]], [1, 1, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 1, 0, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 1, 3, 1, 2, 3, 3, 3, 1, 3, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1 ]], [[[2, 3], [4, 5], [6, 7], [8, 9], [10, 11], [8, 13], [14, 15], [16, 17], [ 18, 19], [20, 21], [22, 23], [16, 17], [26, 27], [28, 29], [30, 31], [16, 32], [33, 34], [16, 35], [36, 37], [38, 39], [40, 41], [42, 43], [44, 45], [16, 32], [33, 34], [49, 50], [51, 52], [53, 54], [55, 56], [57, 58], [59, 15], [60, 61], [62, 63], [64, 65], [66, 67], [68, 69], [70, 71], [38, 72], [73, 74], [53, 76], [77, 78], [62, 80], [81, 82], [20, 21], [83, 84], [60, 61], [62, 63], [64, 65], [49, 91], [92, 93], [62, 63], [96, 97], [53, 76], [77, 78], [101, 102], [103, 104], [68, 106], [107, 108], [109, 110], [62, 63], [64, 65], [62, 115], [116, 117], [49, 50], [118, 119], [62, 63], [64, 65], [68, 120], [121, 122], [16, 35] , [123, 124], [125, 126], [101, 102], [129, 130], [53, 76], [132, 133], [53, 76 ], [132, 133], [62, 115], [137, 138], [68, 69], [139, 140], [42, 43], [44, 45], [62, 63], [64, 65], [62, 115], [116, 117], [49, 50], [118, 119], [146, 147], [ 68, 69], [148, 149], [62, 115], [116, 117], [49, 50], [118, 119], [132, 133], [ 53, 76], [132, 133], [101, 102], [160, 161], [38, 72], [162, 163], [68, 120], [ 165, 166], [68, 69], [139, 140], [53, 54], [55, 56], [62, 115], [116, 117], [49 , 50], [118, 119], [174, 175], [16, 35], [176, 177], [62, 63], [96, 97], [178, 179], [49, 50], [180, 181], [68, 69], [70, 71], [101, 102], [129, 130], [101, 102], [160, 161], [38, 72], [162, 163], [132, 133], [53, 76], [132, 133], [53, 76], [132, 133], [174, 175], [16, 35], [176, 177], [16, 35], [123, 124], [174, 175], [16, 35], [176, 177], [62, 63], [96, 97], [68, 69], [148, 149], [16, 35], [123, 124], [174, 175], [16, 35], [176, 177], [62, 63], [96, 97], [53, 76], [ 132, 133], [53, 76], [132, 133], [160, 161], [38, 72], [162, 163], [101, 102], [129, 130], [178, 179], [49, 50], [180, 181], [16, 35], [123, 124], [174, 175], [16, 35], [176, 177], [62, 63], [96, 97], [16, 35], [176, 177], [68, 69], [70, 71], [49, 50], [180, 181], [62, 63], [96, 97], [38, 72], [162, 163], [38, 72], [162, 163], [101, 102], [129, 130], [53, 76], [132, 133], [16, 35], [176, 177], [68, 69], [70, 71], [16, 35], [176, 177], [68, 69], [70, 71], [16, 35], [123, 124], [16, 35], [176, 177], [68, 69], [70, 71], [38, 72], [162, 163], [101, 102 ], [129, 130], [49, 50], [180, 181], [62, 63], [96, 97], [16, 35], [176, 177], [68, 69], [70, 71], [68, 69], [70, 71], [62, 63], [96, 97]], [1, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 5, 4, 7, 1, 3, 1, 7, 2, 4, 3, 2, 1, 3, 5, 3, 4, 6, 7, 4, 3, 3 , 5, 3, 7, 1, 2, 6, 4, 4, 3, 7, 2, 3, 3, 3, 5, 5, 7, 3, 5, 4, 4, 6, 2, 7, 7, 4, 3, 5, 3, 1, 5, 3, 3, 5, 7, 5, 1, 7, 6, 6, 2, 4, 4, 4, 4, 3, 1, 7, 1, 3, 2, 3, 5 , 3, 1, 5, 3, 7, 7, 1, 3, 1, 5, 3, 4, 4, 4, 6, 2, 2, 6, 7, 5, 7, 1, 4, 6, 3, 1, 5, 3, 1, 1, 7, 3, 5, 5, 5, 3, 7, 1, 6, 2, 6, 2, 2, 6, 4, 4, 4, 4, 4, 1, 1, 7, 1 , 7, 1, 1, 7, 3, 5, 7, 1, 1, 7, 1, 1, 7, 3, 5, 4, 4, 4, 4, 2, 2, 6, 6, 2, 5, 5, 3, 1, 7, 1, 1, 7, 3, 5, 1, 7, 7, 1, 5, 3, 3, 5, 2, 6, 2, 6, 6, 2, 4, 4, 1, 7, 7 , 1, 1, 7, 7, 1, 1, 7, 1, 7, 7, 1, 2, 6, 6, 2, 5, 3, 3, 5, 1, 7, 7, 1, 7, 1, 3, 5]]], [[[[2, 3, 4], [2, 2, 0], [2, 0, 3], [5, 0, 4], [5, 5, 0]], [1, 1, 1, 2, 2 ]], [[[2, 3, 4], [5, 6, 7], [8, 9, 10], [11, 12, 13], [5, 14, 15], [16, 17, 15] , [19, 0, 0], [5, 20, 21], [0, 22, 0], [8, 23, 24], [25, 26, 27], [28, 0, 0], [ 29, 30, 13], [31, 14, 32], [19, 0, 0], [16, 34, 32], [16, 17, 15], [19, 0, 0], [19, 37, 0], [5, 34, 39], [19, 0, 0], [28, 22, 0], [19, 0, 0], [8, 23, 24], [25 , 41, 42], [25, 43, 44], [28, 0, 0], [28, 22, 0], [46, 47, 48], [28, 0, 0], [31 , 17, 39], [19, 0, 0], [19, 37, 0], [5, 34, 39], [19, 0, 0], [19, 37, 0], [19, 37, 0], [5, 34, 39], [19, 0, 0], [28, 22, 0], [54, 41, 55], [28, 0, 0], [25, 43 , 44], [28, 0, 0], [28, 22, 0], [46, 43, 55], [46, 58, 42], [28, 0, 0], [19, 0, 0], [19, 37, 0], [19, 0, 0], [19, 37, 0], [19, 0, 0], [54, 58, 44], [28, 0, 0], [28, 0, 0], [28, 0, 0], [46, 58, 42], [46, 58, 42]], [1, 1, 1, 2, 1, 4, 6, 1, 0 , 1, 2, 3, 8, 7, 6, 4, 4, 6, 6, 1, 6, 3, 6, 1, 2, 2, 3, 3, 8, 3, 7, 6, 6, 1, 6, 6, 6, 1, 6, 3, 5, 3, 2, 3, 3, 8, 8, 3, 6, 6, 6, 6, 6, 5, 3, 3, 3, 8, 8]], [[[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], [14, 41, 42], [43, 44, 45], [46, 47, 48], [49, 50, 51], [52, 53, 54], [46, 56, 57], [58, 59, 60], [61, 0, 62], [63, 0, 64], [14, 65, 66], [67, 68, 69], [70, 71, 72], [73, 74, 0], [75, 76, 77], [78, 0, 79], [80, 81, 25], [ 82, 83, 84], [85, 86, 87], [88, 89, 90], [91, 92, 93], [94, 95, 96], [97, 98, 99], [61, 101, 0], [102, 0, 103], [104, 105, 106], [107, 108, 109], [110, 111, 40], [112, 113, 114], [115, 116, 117], [118, 119, 120], [121, 122, 123], [124, 125, 126], [127, 128, 129], [61, 0, 62], [132, 0, 133], [49, 134, 135], [136, 137, 138], [139, 125, 141], [142, 143, 135], [145, 146, 147], [115, 149, 150], [127, 128, 129], [61, 0, 62], [132, 0, 133], [58, 158, 129], [160, 161, 162], [ 163, 0, 164], [165, 74, 167], [0, 0, 0], [165, 74, 167], [0, 0, 0], [171, 172, 173], [174, 175, 176], [14, 177, 178], [179, 180, 181], [182, 183, 184], [185, 158, 187], [61, 0, 62], [188, 0, 189], [73, 190, 191], [61, 192, 193], [194, 98 , 196], [197, 76, 198], [63, 0, 189], [73, 190, 191], [0, 0, 0], [202, 203, 204 ], [205, 206, 207], [58, 59, 60], [61, 0, 62], [208, 0, 209], [210, 24, 211], [ 82, 83, 84], [212, 213, 214], [88, 215, 216], [217, 218, 219], [220, 221, 222], [223, 224, 216], [226, 227, 228], [229, 230, 231], [232, 233, 196], [235, 0, 193], [237, 0, 164], [97, 233, 240], [241, 242, 77], [188, 0, 133], [165, 74, 167], [235, 101, 62], [73, 190, 191], [0, 0, 0], [250, 251, 252], [253, 254, 255], [256, 257, 258], [232, 233, 196], [235, 0, 193], [102, 0, 103], [264, 265 , 266], [267, 268, 269], [270, 271, 272], [273, 274, 275], [139, 277, 278], [ 127, 128, 129], [61, 0, 62], [132, 0, 133], [118, 284, 272], [286, 287, 288], [ 289, 290, 291], [292, 284, 294], [121, 122, 123], [139, 277, 278], [58, 59, 60] , [61, 0, 62], [63, 0, 64], [127, 59, 187], [300, 301, 302], [303, 0, 304], [ 165, 74, 167], [0, 0, 0], [165, 74, 167], [0, 0, 0], [309, 310, 311], [312, 277 , 126], [314, 315, 42], [316, 317, 318], [289, 320, 321], [58, 59, 60], [61, 0, 62], [63, 0, 64], [142, 50, 322], [323, 324, 325], [312, 277, 126], [142, 143, 135], [145, 146, 147], [115, 149, 150], [127, 128, 129], [61, 0, 62], [132, 0, 133], [127, 59, 187], [300, 301, 302], [303, 0, 304], [165, 74, 167], [0, 0, 0] , [165, 74, 167], [0, 0, 0], [341, 342, 343], [303, 0, 304], [185, 128, 60], [ 160, 161, 162], [303, 0, 304], [73, 190, 191], [0, 0, 0], [165, 74, 167], [61, 192, 193], [0, 0, 0], [165, 74, 167], [61, 192, 193], [0, 0, 0], [292, 284, 294 ], [360, 361, 362], [312, 364, 141], [127, 128, 129], [61, 0, 62], [132, 0, 133 ], [365, 366, 367], [368, 369, 150], [14, 177, 178], [316, 317, 318], [373, 374 , 291], [185, 158, 187], [61, 0, 62], [188, 0, 189], [185, 128, 60], [341, 342, 343], [237, 0, 384], [165, 74, 167], [0, 0, 0], [235, 101, 62], [0, 0, 0], [61, 192, 193], [0, 0, 0], [194, 98, 196], [241, 242, 77], [132, 0, 64], [97, 233, 240], [132, 0, 64], [0, 0, 0], [235, 101, 62], [0, 0, 0], [202, 315, 178], [394 , 395, 396], [397, 398, 399], [314, 315, 42], [402, 403, 404], [405, 406, 407], [165, 74, 167], [0, 0, 0], [314, 409, 410], [411, 412, 413], [210, 24, 211], [ 82, 83, 84], [212, 213, 214], [414, 415, 416], [417, 418, 419], [420, 421, 422] , [423, 424, 425], [426, 427, 428], [194, 429, 240], [235, 0, 193], [303, 0, 384], [223, 434, 435], [436, 437, 438], [417, 418, 419], [223, 224, 216], [442, 443, 444], [445, 446, 447], [232, 233, 196], [235, 0, 193], [237, 0, 164], [232 , 429, 99], [75, 453, 454], [132, 0, 64], [73, 190, 191], [0, 0, 0], [73, 190, 191], [0, 0, 0], [75, 453, 454], [63, 0, 189], [232, 429, 99], [241, 242, 77], [165, 74, 167], [0, 0, 0], [165, 74, 167], [0, 0, 0], [73, 190, 191], [235, 101 , 62], [0, 0, 0], [250, 458, 459], [460, 461, 462], [463, 464, 465], [466, 458, 468], [469, 470, 471], [220, 472, 473], [194, 429, 240], [235, 0, 193], [303, 0 , 384], [232, 429, 99], [75, 453, 454], [132, 0, 64], [73, 190, 191], [0, 0, 0] , [480, 481, 482], [483, 484, 485], [486, 487, 488], [232, 233, 196], [235, 0, 193], [102, 0, 103], [270, 119, 294], [490, 491, 492], [373, 320, 495], [118, 119, 120], [273, 274, 275], [312, 364, 141], [58, 59, 60], [61, 0, 62], [63, 0, 64], [127, 59, 187], [300, 301, 302], [303, 0, 304], [165, 74, 167], [0, 0, 0], [504, 505, 506], [373, 320, 495], [510, 134, 322], [145, 146, 147], [368, 116, 513], [185, 158, 187], [61, 0, 62], [188, 0, 189], [292, 271, 120], [504, 505, 506], [517, 374, 321], [139, 277, 278], [58, 59, 60], [61, 0, 62], [63, 0, 64], [237, 0, 384], [58, 158, 129], [300, 301, 302], [237, 0, 384], [73, 190, 191], [0, 0, 0], [165, 74, 167], [0, 0, 0], [165, 74, 167], [0, 0, 0], [14, 177, 178] , [523, 524, 525], [517, 290, 495], [58, 59, 60], [61, 0, 62], [314, 177, 528], [529, 530, 531], [14, 177, 178], [316, 317, 318], [373, 374, 291], [185, 158, 187], [61, 0, 62], [188, 0, 189], [124, 364, 278], [202, 41, 528], [537, 538, 539], [373, 374, 291], [58, 59, 60], [61, 0, 62], [115, 149, 150], [61, 0, 62], [132, 0, 133], [237, 0, 384], [58, 158, 129], [300, 301, 302], [237, 0, 384], [ 73, 190, 191], [0, 0, 0], [165, 74, 167], [0, 0, 0], [165, 74, 167], [0, 0, 0], [127, 59, 187], [341, 342, 343], [163, 0, 164], [73, 190, 191], [0, 0, 0], [185 , 128, 60], [303, 0, 304], [73, 190, 191], [0, 0, 0], [235, 101, 62], [0, 0, 0] , [0, 0, 0], [61, 192, 193], [0, 0, 0], [61, 192, 193], [0, 0, 0], [292, 271, 120], [504, 505, 506], [517, 374, 321], [270, 271, 272], [360, 361, 362], [124, 125, 126], [58, 59, 60], [61, 0, 62], [292, 284, 294], [360, 361, 362], [312, 364, 141], [127, 128, 129], [61, 0, 62], [14, 177, 178], [316, 317, 318], [373, 374, 291], [185, 158, 187], [61, 0, 62], [188, 0, 189], [185, 128, 60], [237, 0 , 384], [165, 74, 167], [0, 0, 0], [127, 59, 187], [341, 342, 343], [163, 0, 164], [73, 190, 191], [0, 0, 0], [73, 190, 191], [0, 0, 0], [132, 0, 64], [232, 429, 99], [241, 242, 77], [75, 453, 454], [63, 0, 189], [73, 190, 191], [529, 530, 531], [270, 271, 272], [273, 274, 275], [139, 277, 278], [127, 128, 129], [61, 0, 62], [132, 0, 133], [314, 177, 528], [529, 530, 531], [314, 315, 42], [ 537, 538, 539], [517, 290, 495], [185, 158, 187], [61, 0, 62], [188, 0, 189], [ 314, 177, 528], [570, 571, 572], [573, 574, 575], [185, 158, 187], [61, 0, 62], [188, 0, 189], [576, 577, 578], [579, 580, 581], [582, 583, 584], [194, 429, 240], [235, 0, 193], [303, 0, 384], [420, 577, 587], [588, 589, 590], [445, 592 , 593], [576, 577, 578], [423, 424, 425], [598, 599, 600], [97, 98, 99], [235, 0, 193], [163, 0, 304], [197, 76, 198], [63, 0, 189], [73, 190, 191], [73, 190, 191], [0, 0, 0], [601, 602, 603], [604, 605, 606], [607, 608, 587], [610, 611, 612], [598, 599, 600], [194, 429, 240], [235, 0, 193], [303, 0, 384], [223, 224 , 216], [442, 443, 444], [445, 446, 447], [232, 233, 196], [235, 0, 193], [237, 0, 164], [232, 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9, 32, 123], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18 , 8], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9 , 104, 32, 9, 68], [68, 68, 11, 2, 18, 5, 11, 88, 11, 107, 16, 32, 5, 2, 107, 68, 2, 107, 32, 18, 9, 32, 123], [74, 74, 4, 88, 8, 32, 4, 6, 4, 19, 5, 123, 32 , 88, 19, 74, 88, 19, 123, 8, 68, 123, 11], [68, 68, 11, 2, 18, 5, 11, 88, 11, 107, 16, 32, 5, 2, 107, 68, 2, 107, 32, 18, 9, 32, 123], [68, 68, 11, 2, 18, 5, 11, 88, 11, 107, 16, 32, 5, 2, 107, 68, 2, 107, 32, 18, 9, 32, 123], [44, 44, 88, 74, 5, 107, 88, 7, 88, 9, 104, 19, 107, 74, 9, 44, 74, 9, 19, 5, 22, 19, 2] , [68, 68, 11, 2, 18, 5, 11, 88, 11, 107, 16, 32, 5, 2, 107, 68, 2, 107, 32, 18 , 9, 32, 123], [88, 88, 16, 4, 9, 22, 16, 104, 16, 123, 8, 44, 22, 4, 123, 88, 4, 123, 44, 9, 2, 44, 12], [68, 68, 11, 2, 18, 5, 11, 88, 11, 107, 16, 32, 5, 2 , 107, 68, 2, 107, 32, 18, 9, 32, 123], [104, 104, 8, 5, 2, 74, 8, 32, 8, 12, 68, 7, 74, 5, 12, 104, 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107, 32, 18, 9, 32, 123], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8] , [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18 , 88, 104, 18, 8], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16 , 107, 32, 16, 18, 88, 104, 18, 8], [74, 74, 4, 88, 8, 32, 4, 6, 4, 19, 5, 123, 32, 88, 19, 74, 88, 19, 123, 8, 68, 123, 11], [107, 107, 22, 32, 88, 7, 22, 123 , 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8] , [74, 74, 4, 88, 8, 32, 4, 6, 4, 19, 5, 123, 32, 88, 19, 74, 88, 19, 123, 8, 68, 123, 11], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [107, 107, 22, 32, 88, 7, 22, 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7, 2, 4, 7, 18], [104, 104, 8, 5, 2, 74, 8, 32, 8, 12, 68, 7, 74, 5, 12, 104, 5, 12, 7, 2, 4, 7, 18], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8], [107, 107, 22, 32, 88, 7, 22, 123, 22, 16, 74 , 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8], [107, 107, 22, 32, 88, 7, 22 , 123, 22, 16, 74, 18, 7, 32, 16, 107, 32, 16, 18, 88, 104, 18, 8], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68] , [104, 104, 8, 5, 2, 74, 8, 32, 8, 12, 68, 7, 74, 5, 12, 104, 5, 12, 7, 2, 4, 7, 18], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9 , 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12 , 74, 8, 6, 9, 27, 44, 8, 123, 44, 8, 9, 104, 32, 9, 68], [123, 123, 74, 44, 104, 27, 74, 12, 74, 8, 6, 9, 27, 44, 8, 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