Created: Oct. 2013 This is AutoSquared It is one of the packages that accompany the article "A Case Study in Meta-Automation: Automatic Generation of Automata For Prov\ ing Congruence Properties of Combinatorial Sequences" by Eric Rowland and Doron Zeilberger available from Rowland's and Zeilberger's websites and arxiv.org Please report bugs to zeilberg at math dot rutgers dot edu The most current version of this package and paper are available from http://www.math.rutgers.edu/~zeilberg/ . ----------------------------------------------------------------------------\ ------ For a list of the MAIN Auotomata procedures, type, ezraA();, for help with a\ particular procedure, type ezra(ProcedureName); ----------------------------------------------------------------------------\ ------ ----------------------------------------------------------------------------\ ------ For a list of the supporting procedures, type, ezra1();, for help with a par\ ticular procedure, type ezra(ProcedureName); ----------------------------------------------------------------------------\ ------ ----------------------------------------------------------------------------\ ------ For a list of the Congruence automata, for multi-variables, type, ezraAm();, for help with a particular procedure, type: ezra(ProcedureName); ----------------------------------------------------------------------------\ ------ ----------------------------------------------------------------------------\ ------ For a list of the Linear scheme procedures, for one variable, type, ezraL();, for help with a particular procedure, type: ezra(ProcedureName); ----------------------------------------------------------------------------\ ------ ----------------------------------------------------------------------------\ ------ For a list of the multi-variable procedures, for Congruence Linear Schemes type, ezraLm();, for help with a particular procedure, type ezra(ProcedureName); ----------------------------------------------------------------------------\ ------ --------------------------------------------------- All the Linear Schemes whose size is less than, 3000, for determining the, Catalan, numbers modulo prime and prime powers for the first, 1, primes and powers <=, 8 By Shalosh B. Ekhad Recall that the, Catalan, numbers, let's call them C(n), may defined as the constant term of n (x + 2 + 1/x) (1 - x) We are interested in Linear schemes for computing super-fast C(n) modulo pri\ mes, and modulo prime powers allowing Linear schemes with at most, 3000, states. for the first, 8, primes and as high powers as we can afford with our limit of, 3000, states. Thanks to the Chinese Remainder Theorem, this would enable us to determine C\ (n) modulo many composite numbers as well. -------------------------------------- For C(n) modulo, 2, there is a Linear Schemes Scheme with 2, states Here it is: [[[A[2], A[1]], [A[2], 0]], [1, 1]] Just for fun, , C(100000000000000000000), modulo , 2, is , 0 and the next, 10, terms in the sequence C(n) modulo, 2, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 4, there is a Linear Schemes Scheme with 2, states Here it is: [[[A[2], A[1]], [A[2], 2*A[2]]], [1, 1]] Just for fun, , C(100000000000000000000), modulo , 4, is , 0 and the next, 10, terms in the sequence C(n) modulo, 4, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 8, there is a Linear Schemes Scheme with 7, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[2], A[6]], [A[4], A[7]], [2*A[4], 6*A[5]], [A[ 2]+4*A[4], 4*A[2]+A[3]], [6*A[4], 2*A[5]]], [1, 1, 1, 1, 2, 5, 6]] Just for fun, , C(100000000000000000000), modulo , 8, is , 0 and the next, 10, terms in the sequence C(n) modulo, 8, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 16, there is a Linear Schemes Scheme with 17, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[4] , A[12]], [A[13], A[14]], [A[8], A[15]], [A[16], A[17]], [2*A[8], 10*A[9]], [8* A[4]+6*A[10], 8*A[5]+14*A[11]], [8*A[4]+9*A[10], 8*A[5]+9*A[11]], [A[4]+4*A[8], A[5]+8*A[6]+12*A[9]], [12*A[4]+A[6], 4*A[5]+A[7]+8*A[11]], [12*A[4]+8*A[8]+A[10 ], 4*A[5]+8*A[9]+A[11]], [14*A[8], 6*A[9]], [8*A[4]+10*A[10], 8*A[5]+2*A[11]]], [1, 1, 1, 1, 2, 1, 5, 1, 14, 2, 4, 10, 5, 13, 6, 14, 12]] Just for fun, , C(100000000000000000000), modulo , 16, is , 0 and the next, 10, terms in the sequence C(n) modulo, 16, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 32, there is a Linear Schemes Scheme with 37, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12 ], A[13]], [A[14], A[15]], [A[16], A[17]], [A[18], A[19]], [A[20], A[21]], [A[ 22], A[23]], [A[8], A[24]], [A[25], A[26]], [A[27], A[28]], [A[29], A[30]], [A[ 16], A[31]], [A[32], A[33]], [A[34], A[35]], [A[36], A[37]], [2*A[16], 18*A[17] ], [16*A[12]+10*A[18], 16*A[13]+26*A[19]], [24*A[8]+22*A[20], 24*A[9]+6*A[21]], [8*A[10]+30*A[22], 8*A[11]+14*A[23]], [16*A[12]+17*A[18], 16*A[13]+17*A[19]], [ 8*A[8]+16*A[16]+9*A[20], 8*A[9]+16*A[17]+9*A[21]], [24*A[10]+16*A[18]+25*A[22], 24*A[11]+16*A[19]+25*A[23]], [A[8]+4*A[16], A[9]+16*A[14]+20*A[17]], [16*A[8]+A [10]+8*A[12]+28*A[18], 16*A[9]+A[11]+24*A[13]+12*A[19]], [12*A[8]+A[12]+16*A[20 ], 28*A[9]+A[13]], [20*A[10]+16*A[12]+A[14]+8*A[22], 4*A[11]+16*A[13]+A[15]+24* A[23]], [24*A[12]+A[18]+16*A[20], 8*A[13]+A[19]+16*A[21]], [12*A[8]+8*A[16]+A[ 20], 28*A[9]+8*A[17]+A[21]], [4*A[10]+24*A[18]+A[22], 20*A[11]+A[23]+24*A[19]], [14*A[16], 30*A[17]], [16*A[12]+6*A[18], 16*A[13]+22*A[19]], [24*A[8]+26*A[20], 10*A[21]+24*A[9]], [8*A[10]+18*A[22], 8*A[11]+2*A[23]]], [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]] Just for fun, , C(100000000000000000000), modulo , 32, is , 0 and the next, 10, terms in the sequence C(n) modulo, 32, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 64, there is a Linear Schemes Scheme with 77, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12 ], A[13]], [A[14], A[15]], [A[16], A[17]], [A[18], A[19]], [A[20], A[21]], [A[ 22], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A [32], A[33]], [A[34], A[35]], [A[36], A[37]], [A[38], A[39]], [A[40], A[41]], [ A[42], A[43]], [A[44], A[45]], [A[46], A[47]], [A[16], A[48]], [A[49], A[50]], [A[51], A[52]], [A[53], A[54]], [A[55], A[56]], [A[57], A[58]], [A[59], A[60]], [A[61], A[62]], [A[32], A[63]], [A[64], A[65]], [A[66], A[67]], [A[68], A[69]], [A[70], A[71]], [A[72], A[73]], [A[74], A[75]], [A[76], A[77]], [2*A[32], 34*A[ 33]], [32*A[28]+18*A[34], 50*A[35]+32*A[29]], [16*A[24]+42*A[36], 10*A[37]+16*A [25]], [48*A[26]+58*A[38], 26*A[39]+48*A[27]], [32*A[32]+56*A[16]+54*A[40], 32* A[33]+22*A[41]+56*A[17]], [6*A[42]+32*A[34]+24*A[18], 32*A[35]+38*A[43]+24*A[19 ]], [30*A[44]+32*A[36]+8*A[20], 8*A[21]+32*A[37]+62*A[45]], [46*A[46]+32*A[38]+ 40*A[22], 32*A[39]+14*A[47]+40*A[23]], [32*A[28]+33*A[34], 32*A[29]+33*A[35]], [16*A[24]+17*A[36]+32*A[40], 16*A[25]+17*A[37]+32*A[41]], [48*A[26]+49*A[38]+32 *A[42], 48*A[27]+49*A[39]+32*A[43]], [8*A[16]+16*A[32]+9*A[40], 9*A[41]+8*A[17] +16*A[33]], [40*A[18]+16*A[34]+41*A[42], 16*A[35]+40*A[19]+41*A[43]], [25*A[44] +48*A[36]+24*A[20], 24*A[21]+48*A[37]+25*A[45]], [57*A[46]+48*A[38]+56*A[22], 48*A[39]+57*A[47]+56*A[23]], [4*A[32]+A[16], 36*A[33]+A[17]+32*A[30]], [16*A[28 ]+52*A[34]+32*A[20]+A[18], 32*A[21]+20*A[35]+48*A[29]+A[19]], [8*A[24]+16*A[16] +28*A[36]+A[20], A[21]+60*A[37]+16*A[17]+40*A[25]], [24*A[26]+12*A[38]+48*A[18] +A[22], 44*A[39]+56*A[27]+A[23]+48*A[19]], [A[24]+12*A[16]+16*A[40], 48*A[41]+ 44*A[17]+A[25]], [32*A[28]+A[26]+28*A[18], A[27]+32*A[29]+32*A[43]+60*A[19]], [ A[28]+40*A[44]+48*A[24]+52*A[20], 20*A[21]+48*A[25]+A[29]+8*A[45]], [24*A[46]+ 16*A[26]+A[30]+4*A[22], 16*A[27]+56*A[47]+36*A[23]+A[31]], [48*A[28]+A[34]+32*A [44], A[35]+16*A[29]+32*A[45]], [24*A[24]+32*A[16]+A[36]+16*A[40], A[37]+16*A[ 41]+32*A[17]+56*A[25]], [48*A[42]+8*A[26]+A[38]+32*A[18], A[39]+40*A[27]+48*A[ 43]+32*A[19]], [8*A[32]+12*A[16]+A[40], 8*A[33]+A[41]+44*A[17]], [A[42]+40*A[34 ]+60*A[18], 40*A[35]+A[43]+28*A[19]], [A[44]+32*A[24]+24*A[36]+36*A[20], 4*A[21 ]+24*A[37]+32*A[25]+A[45]], [A[46]+32*A[26]+56*A[38]+20*A[22], 56*A[39]+32*A[27 ]+A[47]+52*A[23]], [14*A[32], 46*A[33]], [32*A[28]+62*A[34], 30*A[35]+32*A[29]] , [16*A[24]+38*A[36], 6*A[37]+16*A[25]], [48*A[26]+22*A[38], 54*A[39]+48*A[27]] , [32*A[32]+56*A[16]+26*A[40], 32*A[33]+58*A[41]+56*A[17]], [10*A[42]+32*A[34]+ 24*A[18], 32*A[35]+42*A[43]+24*A[19]], [50*A[44]+32*A[36]+8*A[20], 8*A[21]+32*A [37]+18*A[45]], [34*A[46]+32*A[38]+40*A[22], 32*A[39]+2*A[47]+40*A[23]]], [1, 1 , 1, 1, 2, 1, 5, 1, 14, 2, 4, 1, 42, 5, 45, 1, 22, 14, 12, 2, 28, 4, 8, 1, 62, 42, 52, 5, 34, 45, 61, 1, 38, 22, 60, 14, 36, 12, 24, 2, 44, 28, 24, 4, 56, 8, 16, 54, 62, 44, 42, 12, 52, 40, 5, 22, 34, 36, 45, 18, 61, 29, 6, 38, 28, 22, 20, 60, 56, 14, 52, 36, 40, 12, 40, 24, 48]] Just for fun, , C(100000000000000000000), modulo , 64, is , 0 and the next, 10, terms in the sequence C(n) modulo, 64, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 128, there is a Linear Schemes Scheme with 157, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12 ], A[13]], [A[14], A[15]], [A[16], A[17]], [A[18], A[19]], [A[20], A[21]], [A[ 22], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A [32], A[33]], [A[34], A[35]], [A[36], A[37]], [A[38], A[39]], [A[40], A[41]], [ A[42], A[43]], [A[44], A[45]], [A[46], A[47]], [A[48], A[49]], [A[50], A[51]], [A[52], A[53]], [A[54], A[55]], [A[56], A[57]], [A[58], A[59]], [A[60], A[61]], [A[62], A[63]], [A[64], A[65]], [A[66], A[67]], [A[68], A[69]], [A[70], A[71]], [A[72], A[73]], [A[74], A[75]], [A[76], A[77]], [A[78], A[79]], [A[80], A[81]], [A[82], A[83]], [A[84], A[85]], [A[86], A[87]], [A[88], A[89]], [A[90], A[91]], [A[92], A[93]], [A[94], A[95]], [A[32], A[96]], [A[97], A[98]], [A[99], A[100]] , [A[101], A[102]], [A[103], A[104]], [A[105], A[106]], [A[107], A[108]], [A[ 109], A[110]], [A[111], A[112]], [A[113], A[114]], [A[115], A[116]], [A[117], A [118]], [A[119], A[120]], [A[121], A[122]], [A[123], A[124]], [A[125], A[126]], [A[64], A[127]], [A[128], A[129]], [A[130], A[131]], [A[132], A[133]], [A[134], A[135]], [A[136], A[137]], [A[138], A[139]], [A[140], A[141]], [A[142], A[143]] , [A[144], A[145]], [A[146], A[147]], [A[148], A[149]], [A[150], A[151]], [A[ 152], A[153]], [A[154], A[155]], [A[156], A[157]], [2*A[64], 66*A[65]], [64*A[ 60]+34*A[66], 98*A[67]+64*A[61]], [82*A[68]+32*A[56], 18*A[69]+32*A[57]], [96*A [58]+114*A[70], 96*A[59]+50*A[71]], [64*A[32]+42*A[72]+16*A[48], 64*A[33]+106*A [73]+16*A[49]], [74*A[74]+64*A[34]+80*A[50], 64*A[35]+10*A[75]+80*A[51]], [48*A [52]+122*A[76]+64*A[36], 64*A[37]+48*A[53]+58*A[77]], [112*A[54]+26*A[78]+64*A[ 38], 64*A[39]+90*A[79]+112*A[55]], [120*A[32]+54*A[80]+32*A[64], 32*A[65]+120*A [33]+118*A[81]], [56*A[34]+32*A[66]+86*A[82], 32*A[67]+56*A[35]+22*A[83]], [6*A [84]+32*A[68]+24*A[36], 32*A[69]+24*A[37]+70*A[85]], [32*A[70]+38*A[86]+88*A[38 ], 88*A[39]+32*A[71]+102*A[87]], [94*A[88]+32*A[72]+64*A[48]+72*A[40], 72*A[41] +30*A[89]+32*A[73]+64*A[49]], [8*A[42]+32*A[74]+126*A[90]+64*A[50], 32*A[75]+64 *A[51]+8*A[43]+62*A[91]], [64*A[52]+32*A[76]+104*A[44]+46*A[92], 64*A[53]+32*A[ 77]+104*A[45]+110*A[93]], [40*A[46]+64*A[54]+78*A[94]+32*A[78], 32*A[79]+40*A[ 47]+64*A[55]+14*A[95]], [64*A[60]+65*A[66], 64*A[61]+65*A[67]], [32*A[56]+33*A[ 68]+64*A[88], 32*A[57]+33*A[69]+64*A[89]], [96*A[58]+97*A[70]+64*A[90], 96*A[59 ]+97*A[71]+64*A[91]], [64*A[32]+16*A[48]+17*A[72]+32*A[80], 64*A[33]+32*A[81]+ 17*A[73]+16*A[49]], [81*A[74]+64*A[34]+80*A[50]+32*A[82], 64*A[35]+32*A[83]+81* A[75]+80*A[51]], [96*A[84]+48*A[52]+49*A[76]+64*A[36], 64*A[37]+48*A[53]+49*A[ 77]+96*A[85]], [112*A[54]+113*A[78]+96*A[86]+64*A[38], 64*A[39]+113*A[79]+112*A [55]+96*A[87]], [8*A[32]+9*A[80]+16*A[64], 16*A[65]+8*A[33]+9*A[81]], [72*A[34] +16*A[66]+73*A[82], 16*A[67]+72*A[35]+73*A[83]], [41*A[84]+80*A[68]+40*A[36], 80*A[69]+40*A[37]+41*A[85]], [80*A[70]+105*A[86]+104*A[38], 104*A[39]+80*A[71]+ 105*A[87]], [25*A[88]+48*A[72]+64*A[48]+24*A[40], 24*A[41]+25*A[89]+48*A[73]+64 *A[49]], [88*A[42]+48*A[74]+89*A[90]+64*A[50], 48*A[75]+64*A[51]+88*A[43]+89*A[ 91]], [64*A[52]+112*A[76]+56*A[44]+57*A[92], 64*A[53]+112*A[77]+56*A[45]+57*A[ 93]], [120*A[46]+64*A[54]+121*A[94]+112*A[78], 112*A[79]+120*A[47]+64*A[55]+121 *A[95]], [A[32]+4*A[64], 68*A[65]+A[33]+64*A[62]], [A[34]+32*A[60]+64*A[44]+100 *A[66], 36*A[67]+A[35]+64*A[45]+96*A[61]], [52*A[68]+80*A[56]+A[36]+96*A[40], 116*A[69]+A[37]+96*A[41]+16*A[57]], [32*A[42]+112*A[58]+20*A[70]+A[38], A[39]+ 48*A[59]+84*A[71]+32*A[43]], [80*A[32]+64*A[80]+28*A[72]+72*A[48]+A[40], 80*A[ 33]+A[41]+64*A[81]+92*A[73]+8*A[49]], [A[42]+124*A[74]+16*A[34]+104*A[50]+64*A[ 82], 16*A[35]+64*A[83]+60*A[75]+40*A[51]+A[43]], [64*A[84]+24*A[52]+76*A[76]+A[ 44]+48*A[36], 48*A[37]+88*A[53]+12*A[77]+64*A[85]+A[45]], [A[46]+56*A[54]+44*A[ 78]+64*A[86]+112*A[38], 112*A[39]+108*A[79]+A[47]+120*A[55]+64*A[87]], [76*A[32 ]+16*A[80]+A[48]+64*A[64], 64*A[65]+12*A[33]+80*A[81]+A[49]], [108*A[34]+A[50]+ 64*A[60]+64*A[66]+112*A[82], 64*A[67]+44*A[35]+48*A[83]+A[51]+64*A[61]], [64*A[ 84]+64*A[68]+32*A[56]+A[52]+92*A[36], 64*A[69]+28*A[37]+A[53]+32*A[57]], [96*A[ 58]+A[54]+64*A[70]+32*A[86]+124*A[38], 60*A[39]+96*A[59]+A[55]+64*A[71]+96*A[87 ]], [40*A[88]+A[56]+64*A[32]+48*A[48]+116*A[40], 64*A[33]+52*A[41]+A[57]+104*A[ 89]+48*A[49]], [20*A[42]+64*A[34]+8*A[90]+A[58]+112*A[50], 64*A[35]+A[59]+112*A [51]+84*A[43]+72*A[91]], [80*A[52]+A[60]+4*A[44]+64*A[36]+88*A[92], 64*A[37]+80 *A[53]+68*A[45]+A[61]+24*A[93]], [36*A[46]+16*A[54]+A[62]+56*A[94]+64*A[38], 64 *A[39]+100*A[47]+16*A[55]+120*A[95]+A[63]], [96*A[60]+A[66]+64*A[92], A[67]+32* A[61]+64*A[93]], [A[68]+32*A[88]+48*A[56]+64*A[40], A[69]+64*A[41]+112*A[57]+32 *A[89]], [64*A[42]+96*A[90]+16*A[58]+A[70], 80*A[59]+A[71]+64*A[43]+96*A[91]], [32*A[32]+16*A[80]+A[72]+24*A[48], 32*A[33]+16*A[81]+A[73]+88*A[49]], [A[74]+32 *A[34]+120*A[50]+80*A[82], 32*A[35]+80*A[83]+A[75]+56*A[51]], [48*A[84]+72*A[52 ]+A[76]+96*A[36], 96*A[37]+8*A[53]+A[77]+48*A[85]], [40*A[54]+A[78]+112*A[86]+ 96*A[38], 96*A[39]+A[79]+104*A[55]+112*A[87]], [12*A[32]+A[80]+8*A[64], 8*A[65] +76*A[33]+A[81]], [108*A[34]+72*A[66]+A[82], 72*A[67]+44*A[35]+A[83]], [A[84]+ 40*A[68]+64*A[56]+60*A[36], 40*A[69]+124*A[37]+A[85]+64*A[57]], [64*A[58]+104*A [70]+A[86]+28*A[38], 92*A[39]+64*A[59]+104*A[71]+A[87]], [A[88]+64*A[32]+24*A[ 72]+32*A[48]+36*A[40], 64*A[33]+100*A[41]+A[89]+24*A[73]+32*A[49]], [4*A[42]+88 *A[74]+64*A[34]+A[90]+32*A[50], 64*A[35]+88*A[75]+32*A[51]+68*A[43]+A[91]], [96 *A[52]+56*A[76]+84*A[44]+64*A[36]+A[92], 64*A[37]+96*A[53]+56*A[77]+20*A[45]+A[ 93]], [52*A[46]+96*A[54]+A[94]+120*A[78]+64*A[38], 64*A[39]+120*A[79]+116*A[47] +96*A[55]+A[95]], [14*A[64], 78*A[65]], [64*A[60]+110*A[66], 46*A[67]+64*A[61]] , [62*A[68]+32*A[56], 126*A[69]+32*A[57]], [96*A[58]+30*A[70], 96*A[59]+94*A[71 ]], [64*A[32]+102*A[72]+16*A[48], 64*A[33]+38*A[73]+16*A[49]], [70*A[74]+64*A[ 34]+80*A[50], 64*A[35]+6*A[75]+80*A[51]], [48*A[52]+22*A[76]+64*A[36], 64*A[37] +48*A[53]+86*A[77]], [112*A[54]+118*A[78]+64*A[38], 64*A[39]+54*A[79]+112*A[55] ], [56*A[32]+26*A[80]+32*A[64], 32*A[65]+56*A[33]+90*A[81]], [120*A[34]+32*A[66 ]+122*A[82], 32*A[67]+120*A[35]+58*A[83]], [74*A[84]+32*A[68]+88*A[36], 32*A[69 ]+88*A[37]+10*A[85]], [32*A[70]+42*A[86]+24*A[38], 24*A[39]+32*A[71]+106*A[87]] , [114*A[88]+32*A[72]+64*A[48]+8*A[40], 8*A[41]+50*A[89]+32*A[73]+64*A[49]], [ 72*A[42]+32*A[74]+82*A[90]+64*A[50], 32*A[75]+64*A[51]+72*A[43]+18*A[91]], [64* A[52]+32*A[76]+40*A[44]+34*A[92], 64*A[53]+32*A[77]+40*A[45]+98*A[93]], [104*A[ 46]+64*A[54]+2*A[94]+32*A[78], 32*A[79]+104*A[47]+64*A[55]+66*A[95]]], [1, 1, 1 , 1, 2, 1, 5, 1, 14, 2, 4, 1, 42, 5, 45, 1, 22, 14, 12, 2, 28, 4, 8, 1, 126, 42 , 116, 5, 34, 45, 125, 1, 102, 22, 124, 14, 100, 12, 88, 2, 44, 28, 24, 4, 56, 8, 16, 1, 118, 126, 44, 42, 12, 116, 104, 5, 86, 34, 36, 45, 82, 125, 29, 1, 6, 102, 92, 22, 20, 124, 56, 14, 116, 100, 104, 12, 40, 88, 48, 2, 76, 44, 120, 28 , 72, 24, 48, 4, 88, 56, 48, 8, 112, 16, 32, 38, 118, 60, 126, 4, 44, 24, 42, 28, 12, 120, 116, 88, 104, 80, 5, 14, 86, 28, 34, 28, 36, 72, 45, 70, 82, 4, 125, 50, 29, 93, 70, 6, 28, 102, 116, 92, 120, 22, 36, 20, 72, 124, 72, 56, 112 , 14, 20, 116, 72, 100, 56, 104, 80, 12, 8, 40, 16, 88, 80, 48, 96]] Just for fun, , C(100000000000000000000), modulo , 128, is , 0 and the next, 10, terms in the sequence C(n) modulo, 128, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 256, there is a Linear Schemes Scheme with 317, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[8], A[9]], [A[10], A[11]], [A[12 ], A[13]], [A[14], A[15]], [A[16], A[17]], [A[18], A[19]], [A[20], A[21]], [A[ 22], A[23]], [A[24], A[25]], [A[26], A[27]], [A[28], A[29]], [A[30], A[31]], [A [32], A[33]], [A[34], A[35]], [A[36], A[37]], [A[38], A[39]], [A[40], A[41]], [ A[42], A[43]], [A[44], A[45]], [A[46], A[47]], [A[48], A[49]], [A[50], A[51]], [A[52], A[53]], [A[54], A[55]], [A[56], A[57]], [A[58], A[59]], [A[60], A[61]], [A[62], A[63]], [A[64], A[65]], [A[66], A[67]], [A[68], A[69]], [A[70], A[71]], [A[72], A[73]], [A[74], A[75]], [A[76], A[77]], [A[78], A[79]], [A[80], A[81]], [A[82], A[83]], [A[84], A[85]], [A[86], A[87]], [A[88], A[89]], [A[90], A[91]], [A[92], A[93]], [A[94], A[95]], [A[96], A[97]], [A[98], A[99]], [A[100], A[101] ], [A[102], A[103]], [A[104], A[105]], [A[106], A[107]], [A[108], A[109]], [A[ 110], A[111]], [A[112], A[113]], [A[114], A[115]], [A[116], A[117]], [A[118], A [119]], [A[120], A[121]], [A[122], A[123]], [A[124], A[125]], [A[126], A[127]], [A[128], A[129]], [A[130], A[131]], [A[132], A[133]], [A[134], A[135]], [A[136] , A[137]], [A[138], A[139]], [A[140], A[141]], [A[142], A[143]], [A[144], A[145 ]], [A[146], A[147]], [A[148], A[149]], [A[150], A[151]], [A[152], A[153]], [A[ 154], A[155]], [A[156], A[157]], [A[158], A[159]], [A[160], A[161]], [A[162], A [163]], [A[164], A[165]], [A[166], A[167]], [A[168], A[169]], [A[170], A[171]], [A[172], A[173]], [A[174], A[175]], [A[176], A[177]], [A[178], A[179]], [A[180] , A[181]], [A[182], A[183]], [A[184], A[185]], [A[186], A[187]], [A[188], A[189 ]], [A[190], A[191]], [A[64], A[192]], [A[193], A[194]], [A[195], A[196]], [A[ 197], A[198]], [A[199], A[200]], [A[201], A[202]], [A[203], A[204]], [A[205], A [206]], [A[207], A[208]], [A[209], A[210]], [A[211], A[212]], [A[213], A[214]], [A[215], A[216]], [A[217], A[218]], [A[219], A[220]], [A[221], A[222]], [A[223] , A[224]], [A[225], A[226]], [A[227], A[228]], [A[229], A[230]], [A[231], A[232 ]], [A[233], A[234]], [A[235], A[236]], [A[237], A[238]], [A[239], A[240]], [A[ 241], A[242]], [A[243], A[244]], [A[245], A[246]], [A[247], A[248]], [A[249], A [250]], [A[251], A[252]], [A[253], A[254]], [A[128], A[255]], [A[256], A[257]], [A[258], A[259]], [A[260], A[261]], [A[262], A[263]], [A[264], A[265]], [A[266] , A[267]], [A[268], A[269]], [A[270], A[271]], [A[272], A[273]], [A[274], A[275 ]], [A[276], A[277]], [A[278], A[279]], [A[280], A[281]], [A[282], A[283]], [A[ 284], A[285]], [A[286], A[287]], [A[288], A[289]], [A[290], A[291]], [A[292], A [293]], [A[294], A[295]], [A[296], A[297]], [A[298], A[299]], [A[300], A[301]], [A[302], A[303]], [A[304], A[305]], [A[306], A[307]], [A[308], A[309]], [A[310] , A[311]], [A[312], A[313]], [A[314], A[315]], [A[316], A[317]], [2*A[128], 130 *A[129]], [128*A[124]+66*A[130], 128*A[125]+194*A[131]], [64*A[120]+162*A[132], 64*A[121]+34*A[133]], [192*A[122]+226*A[134], 192*A[123]+98*A[135]], [82*A[136] +160*A[112]+128*A[80], 128*A[81]+210*A[137]+160*A[113]], [146*A[138]+32*A[114]+ 128*A[82], 128*A[83]+32*A[115]+18*A[139]], [128*A[84]+224*A[116]+242*A[140], 128*A[85]+224*A[117]+114*A[141]], [50*A[142]+128*A[86]+96*A[118], 96*A[119]+128 *A[87]+178*A[143]], [16*A[96]+170*A[144]+128*A[160]+64*A[64], 64*A[65]+128*A[ 161]+16*A[97]+42*A[145]], [144*A[98]+234*A[146]+128*A[162]+64*A[66], 64*A[67]+ 144*A[99]+106*A[147]+128*A[163]], [64*A[68]+74*A[148]+128*A[164]+80*A[100], 64* A[69]+80*A[101]+128*A[165]+202*A[149]], [128*A[166]+138*A[150]+208*A[102]+64*A[ 70], 10*A[151]+208*A[103]+64*A[71]+128*A[167]], [128*A[168]+250*A[152]+192*A[72 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222*A[177]+128*A[65]+192*A[97]+72*A[81]+160* A[145]], [192*A[98]+160*A[146]+158*A[178]+128*A[66]+200*A[82], 128*A[67]+30*A[ 179]+192*A[99]+160*A[147]+200*A[83]], [254*A[180]+136*A[84]+128*A[68]+160*A[148 ]+192*A[100], 128*A[69]+136*A[85]+192*A[101]+126*A[181]+160*A[149]], [160*A[150 ]+192*A[102]+128*A[70]+62*A[182]+8*A[86], 190*A[183]+160*A[151]+192*A[103]+128* A[71]+8*A[87]], [174*A[184]+104*A[88]+160*A[152]+128*A[72]+64*A[104], 46*A[185] +160*A[153]+64*A[105]+104*A[89]+128*A[73]], [128*A[74]+232*A[90]+160*A[154]+238 *A[186]+64*A[106], 160*A[155]+110*A[187]+128*A[75]+64*A[107]+232*A[91]], [78*A[ 188]+160*A[156]+64*A[108]+128*A[76]+168*A[92], 128*A[77]+160*A[157]+206*A[189]+ 64*A[109]+168*A[93]], [40*A[94]+128*A[78]+142*A[190]+160*A[158]+64*A[110], 128* A[79]+160*A[159]+14*A[191]+64*A[111]+40*A[95]], [128*A[124]+129*A[130], 128*A[ 125]+129*A[131]], [64*A[120]+65*A[132]+128*A[184], 64*A[121]+65*A[133]+128*A[ 185]], [192*A[122]+193*A[134]+128*A[186], 192*A[123]+193*A[135]+128*A[187]], [ 128*A[80]+32*A[112]+33*A[136]+64*A[176], 128*A[81]+32*A[113]+33*A[137]+64*A[177 ]], [161*A[138]+64*A[178]+160*A[114]+128*A[82], 64*A[179]+128*A[83]+160*A[115]+ 161*A[139]], [192*A[180]+128*A[84]+96*A[116]+97*A[140], 128*A[85]+192*A[181]+96 *A[117]+97*A[141]], [225*A[142]+192*A[182]+128*A[86]+224*A[118], 192*A[183]+224 *A[119]+128*A[87]+225*A[143]], [16*A[96]+17*A[144]+32*A[160]+192*A[64], 192*A[ 65]+32*A[161]+16*A[97]+17*A[145]], [144*A[98]+145*A[146]+32*A[162]+192*A[66], 192*A[67]+144*A[99]+145*A[147]+32*A[163]], [192*A[68]+81*A[148]+160*A[164]+80*A [100], 192*A[69]+80*A[101]+160*A[165]+81*A[149]], [160*A[166]+209*A[150]+208*A[ 102]+192*A[70], 209*A[151]+208*A[103]+192*A[71]+160*A[167]], [96*A[168]+49*A[ 152]+64*A[72]+48*A[104], 49*A[153]+48*A[105]+64*A[73]+96*A[169]], [96*A[170]+64 *A[74]+177*A[154]+176*A[106], 177*A[155]+96*A[171]+64*A[75]+176*A[107]], [224*A [172]+113*A[156]+112*A[108]+64*A[76], 64*A[77]+224*A[173]+113*A[157]+112*A[109] ], [224*A[174]+64*A[78]+241*A[158]+240*A[110], 224*A[175]+64*A[79]+241*A[159]+ 240*A[111]], [16*A[128]+9*A[160]+8*A[64], 8*A[65]+9*A[161]+16*A[129]], [16*A[ 130]+137*A[162]+136*A[66], 136*A[67]+16*A[131]+137*A[163]], [72*A[68]+144*A[132 ]+73*A[164], 72*A[69]+144*A[133]+73*A[165]], [201*A[166]+200*A[70]+144*A[134], 200*A[71]+201*A[167]+144*A[135]], [41*A[168]+80*A[136]+128*A[112]+40*A[72], 80* A[137]+40*A[73]+128*A[113]+41*A[169]], [169*A[170]+168*A[74]+80*A[138]+128*A[ 114], 169*A[171]+168*A[75]+128*A[115]+80*A[139]], [105*A[172]+104*A[76]+128*A[ 116]+208*A[140], 104*A[77]+105*A[173]+128*A[117]+208*A[141]], [233*A[174]+208*A [142]+232*A[78]+128*A[118], 233*A[175]+232*A[79]+128*A[119]+208*A[143]], [192*A [96]+48*A[144]+24*A[80]+25*A[176]+128*A[64], 25*A[177]+128*A[65]+192*A[97]+24*A [81]+48*A[145]], [192*A[98]+48*A[146]+153*A[178]+128*A[66]+152*A[82], 128*A[67] +153*A[179]+192*A[99]+48*A[147]+152*A[83]], [89*A[180]+88*A[84]+128*A[68]+176*A [148]+192*A[100], 128*A[69]+88*A[85]+192*A[101]+89*A[181]+176*A[149]], [176*A[ 150]+192*A[102]+128*A[70]+217*A[182]+216*A[86], 217*A[183]+176*A[151]+192*A[103 ]+128*A[71]+216*A[87]], [57*A[184]+56*A[88]+112*A[152]+128*A[72]+64*A[104], 57* A[185]+112*A[153]+64*A[105]+56*A[89]+128*A[73]], [128*A[74]+184*A[90]+112*A[154 ]+185*A[186]+64*A[106], 112*A[155]+185*A[187]+128*A[75]+64*A[107]+184*A[91]], [ 121*A[188]+240*A[156]+64*A[108]+128*A[76]+120*A[92], 128*A[77]+240*A[157]+121*A [189]+64*A[109]+120*A[93]], [248*A[94]+128*A[78]+249*A[190]+240*A[158]+64*A[110 ], 128*A[79]+240*A[159]+249*A[191]+64*A[111]+248*A[95]], [4*A[128]+A[64], A[65] +128*A[126]+132*A[129]], [64*A[124]+196*A[130]+A[66]+128*A[92], A[67]+192*A[125 ]+68*A[131]+128*A[93]], [A[68]+192*A[88]+160*A[120]+100*A[132], A[69]+32*A[121] +228*A[133]+192*A[89]], [64*A[90]+A[70]+224*A[122]+36*A[134], 96*A[123]+A[71]+ 164*A[135]+64*A[91]], [52*A[136]+80*A[112]+224*A[80]+A[72]+128*A[176], 128*A[ 177]+224*A[81]+180*A[137]+A[73]+208*A[113]], [A[74]+244*A[138]+128*A[178]+144*A [114]+96*A[82], 128*A[179]+96*A[83]+A[75]+16*A[115]+116*A[139]], [128*A[180]+ 160*A[84]+A[76]+240*A[116]+148*A[140], A[77]+160*A[85]+128*A[181]+112*A[117]+20 *A[141]], [84*A[142]+A[78]+128*A[182]+32*A[86]+48*A[118], 128*A[183]+A[79]+176* A[119]+32*A[87]+212*A[143]], [200*A[96]+28*A[144]+192*A[160]+A[80]+208*A[64], 208*A[65]+192*A[161]+72*A[97]+A[81]+156*A[145]], [8*A[98]+220*A[146]+192*A[162] +80*A[66]+A[82], 80*A[67]+136*A[99]+92*A[147]+A[83]+192*A[163]], [A[84]+144*A[ 68]+124*A[148]+192*A[164]+104*A[100], 144*A[69]+A[85]+232*A[101]+192*A[165]+252 *A[149]], [192*A[166]+60*A[150]+168*A[102]+16*A[70]+A[86], 188*A[151]+40*A[103] +16*A[71]+A[87]+192*A[167]], [64*A[168]+A[88]+76*A[152]+48*A[72]+152*A[104], 204*A[153]+24*A[105]+A[89]+48*A[73]+64*A[169]], [64*A[170]+176*A[74]+A[90]+12*A [154]+216*A[106], 140*A[155]+64*A[171]+176*A[75]+88*A[107]+A[91]], [64*A[172]+ 172*A[156]+56*A[108]+240*A[76]+A[92], 240*A[77]+64*A[173]+44*A[157]+184*A[109]+ A[93]], [64*A[174]+A[94]+112*A[78]+108*A[158]+120*A[110], 64*A[175]+112*A[79]+ 236*A[159]+248*A[111]+A[95]], [A[96]+64*A[128]+16*A[160]+76*A[64], 204*A[65]+ 144*A[161]+A[97]+64*A[129]], [A[98]+128*A[124]+64*A[130]+208*A[162]+140*A[66], 12*A[67]+A[99]+128*A[125]+64*A[131]+80*A[163]], [108*A[68]+64*A[120]+64*A[132]+ 112*A[164]+A[100], 236*A[69]+A[101]+64*A[121]+64*A[133]+240*A[165]], [48*A[166] +A[102]+172*A[70]+192*A[122]+64*A[134], A[103]+192*A[123]+44*A[71]+176*A[167]+ 64*A[135]], [64*A[168]+192*A[136]+160*A[112]+128*A[80]+92*A[72]+A[104], 128*A[ 81]+192*A[137]+A[105]+220*A[73]+160*A[113]+192*A[169]], [156*A[74]+192*A[138]+ 32*A[114]+128*A[82]+A[106], 128*A[83]+128*A[171]+28*A[75]+A[107]+32*A[115]+192* A[139]], [160*A[172]+128*A[84]+A[108]+124*A[76]+224*A[116]+192*A[140], 252*A[77 ]+128*A[85]+32*A[173]+A[109]+224*A[117]+192*A[141]], [96*A[174]+192*A[142]+188* A[78]+128*A[86]+A[110]+96*A[118], 224*A[175]+60*A[79]+96*A[119]+128*A[87]+A[111 ]+192*A[143]], [48*A[96]+A[112]+116*A[80]+40*A[176]+192*A[64], 168*A[177]+192*A [65]+48*A[97]+244*A[81]+A[113]], [176*A[98]+232*A[178]+A[114]+192*A[66]+180*A[ 82], 192*A[67]+104*A[179]+176*A[99]+52*A[83]+A[115]], [136*A[180]+148*A[84]+192 *A[68]+A[116]+112*A[100], 192*A[69]+20*A[85]+112*A[101]+8*A[181]+A[117]], [240* A[102]+192*A[70]+72*A[182]+212*A[86]+A[118], 200*A[183]+240*A[103]+A[119]+192*A [71]+84*A[87]], [88*A[184]+4*A[88]+A[120]+128*A[152]+64*A[72]+80*A[104], 216*A[ 185]+128*A[153]+A[121]+80*A[105]+132*A[89]+64*A[73]], [64*A[74]+68*A[90]+A[122] +128*A[154]+24*A[186]+208*A[106], 128*A[155]+152*A[187]+64*A[75]+208*A[107]+A[ 123]+196*A[91]], [184*A[188]+128*A[156]+144*A[108]+64*A[76]+A[124]+36*A[92], 64 *A[77]+A[125]+128*A[157]+56*A[189]+144*A[109]+164*A[93]], [100*A[94]+64*A[78]+A [126]+120*A[190]+128*A[158]+16*A[110], 64*A[79]+A[127]+128*A[159]+248*A[191]+16 *A[111]+228*A[95]], [128*A[188]+192*A[124]+A[130], 64*A[125]+A[131]+128*A[189]] , [64*A[184]+128*A[88]+96*A[120]+A[132], 64*A[185]+224*A[121]+A[133]+128*A[89]] , [128*A[90]+32*A[122]+192*A[186]+A[134], 192*A[187]+160*A[123]+A[135]+128*A[91 ]], [A[136]+48*A[112]+192*A[80]+32*A[176], 32*A[177]+192*A[81]+A[137]+176*A[113 ]], [A[138]+160*A[178]+240*A[114]+192*A[82], 160*A[179]+192*A[83]+112*A[115]+A[ 139]], [96*A[180]+64*A[84]+144*A[116]+A[140], 64*A[85]+96*A[181]+16*A[117]+A[ 141]], [A[142]+224*A[182]+64*A[86]+80*A[118], 224*A[183]+208*A[119]+64*A[87]+A[ 143]], [24*A[96]+128*A[128]+A[144]+16*A[160]+32*A[64], 32*A[65]+16*A[161]+152*A [97]+128*A[129]+A[145]], [216*A[98]+A[146]+128*A[130]+144*A[162]+32*A[66], 32*A [67]+88*A[99]+A[147]+128*A[131]+144*A[163]], [160*A[68]+A[148]+128*A[132]+80*A[ 164]+120*A[100], 160*A[69]+248*A[101]+128*A[133]+80*A[165]+A[149]], [208*A[166] +A[150]+56*A[102]+160*A[70]+128*A[134], A[151]+184*A[103]+160*A[71]+208*A[167]+ 128*A[135]], [48*A[168]+A[152]+128*A[136]+224*A[72]+72*A[104], A[153]+128*A[137 ]+200*A[105]+224*A[73]+48*A[169]], [176*A[170]+224*A[74]+128*A[138]+A[154]+8*A[ 106], A[155]+176*A[171]+224*A[75]+136*A[107]+128*A[139]], [112*A[172]+A[156]+ 168*A[108]+96*A[76]+128*A[140], 96*A[77]+112*A[173]+A[157]+40*A[109]+128*A[141] ], [240*A[174]+128*A[142]+96*A[78]+A[158]+104*A[110], 240*A[175]+96*A[79]+A[159 ]+232*A[111]+128*A[143]], [136*A[128]+A[160]+12*A[64], 140*A[65]+A[161]+136*A[ 129]], [8*A[130]+A[162]+204*A[66], 76*A[67]+8*A[131]+A[163]], [108*A[68]+128*A[ 120]+200*A[132]+A[164], 236*A[69]+128*A[121]+200*A[133]+A[165]], [A[166]+44*A[ 70]+128*A[122]+72*A[134], 128*A[123]+172*A[71]+A[167]+72*A[135]], [A[168]+168*A [136]+192*A[112]+128*A[80]+60*A[72], 128*A[81]+168*A[137]+188*A[73]+192*A[113]+ A[169]], [A[170]+252*A[74]+40*A[138]+192*A[114]+128*A[82], 128*A[83]+A[171]+124 *A[75]+192*A[115]+40*A[139]], [A[172]+128*A[84]+156*A[76]+64*A[116]+232*A[140], 28*A[77]+128*A[85]+A[173]+64*A[117]+232*A[141]], [A[174]+104*A[142]+92*A[78]+ 128*A[86]+64*A[118], A[175]+220*A[79]+64*A[119]+128*A[87]+104*A[143]], [32*A[96 ]+152*A[144]+128*A[160]+36*A[80]+A[176]+64*A[64], A[177]+64*A[65]+128*A[161]+32 *A[97]+164*A[81]+152*A[145]], [32*A[98]+24*A[146]+A[178]+128*A[162]+64*A[66]+ 228*A[82], 64*A[67]+A[179]+32*A[99]+24*A[147]+100*A[83]+128*A[163]], [A[180]+ 132*A[84]+64*A[68]+216*A[148]+128*A[164]+160*A[100], 64*A[69]+4*A[85]+160*A[101 ]+A[181]+128*A[165]+216*A[149]], [128*A[166]+88*A[150]+160*A[102]+64*A[70]+A[ 182]+68*A[86], A[183]+88*A[151]+160*A[103]+64*A[71]+196*A[87]+128*A[167]], [A[ 184]+128*A[168]+84*A[88]+184*A[152]+192*A[72]+224*A[104], A[185]+184*A[153]+224 *A[105]+212*A[89]+192*A[73]+128*A[169]], [128*A[170]+192*A[74]+20*A[90]+56*A[ 154]+A[186]+224*A[106], 56*A[155]+A[187]+128*A[171]+192*A[75]+224*A[107]+148*A[ 91]], [128*A[172]+A[188]+248*A[156]+96*A[108]+192*A[76]+180*A[92], 192*A[77]+ 128*A[173]+248*A[157]+A[189]+96*A[109]+52*A[93]], [128*A[174]+116*A[94]+192*A[ 78]+A[190]+120*A[158]+96*A[110], 128*A[175]+192*A[79]+120*A[159]+A[191]+96*A[ 111]+244*A[95]], [14*A[128], 142*A[129]], [128*A[124]+206*A[130], 128*A[125]+78 *A[131]], [64*A[120]+110*A[132], 64*A[121]+238*A[133]], [192*A[122]+46*A[134], 192*A[123]+174*A[135]], [190*A[136]+160*A[112]+128*A[80], 128*A[81]+62*A[137]+ 160*A[113]], [126*A[138]+32*A[114]+128*A[82], 128*A[83]+32*A[115]+254*A[139]], [128*A[84]+224*A[116]+30*A[140], 128*A[85]+224*A[117]+158*A[141]], [222*A[142]+ 128*A[86]+96*A[118], 96*A[119]+128*A[87]+94*A[143]], [144*A[96]+102*A[144]+128* A[160]+64*A[64], 64*A[65]+128*A[161]+144*A[97]+230*A[145]], [16*A[98]+38*A[146] +128*A[162]+64*A[66], 64*A[67]+16*A[99]+166*A[147]+128*A[163]], [64*A[68]+198*A [148]+128*A[164]+208*A[100], 64*A[69]+208*A[101]+128*A[165]+70*A[149]], [128*A[ 166]+134*A[150]+80*A[102]+64*A[70], 6*A[151]+80*A[103]+64*A[71]+128*A[167]], [ 128*A[168]+22*A[152]+192*A[72]+48*A[104], 150*A[153]+48*A[105]+192*A[73]+128*A[ 169]], [128*A[170]+192*A[74]+214*A[154]+176*A[106], 86*A[155]+128*A[171]+192*A[ 75]+176*A[107]], [128*A[172]+118*A[156]+112*A[108]+192*A[76], 192*A[77]+128*A[ 173]+246*A[157]+112*A[109]], [128*A[174]+192*A[78]+54*A[158]+240*A[110], 128*A[ 175]+192*A[79]+182*A[159]+240*A[111]], [160*A[128]+154*A[160]+184*A[64], 184*A[ 65]+26*A[161]+160*A[129]], [160*A[130]+90*A[162]+56*A[66], 56*A[67]+160*A[131]+ 218*A[163]], [248*A[68]+160*A[132]+250*A[164], 248*A[69]+160*A[133]+122*A[165]] , [186*A[166]+120*A[70]+160*A[134], 120*A[71]+58*A[167]+160*A[135]], [74*A[168] +160*A[136]+128*A[112]+216*A[72], 160*A[137]+216*A[73]+128*A[113]+202*A[169]], [10*A[170]+88*A[74]+160*A[138]+128*A[114], 138*A[171]+88*A[75]+128*A[115]+160*A [139]], [170*A[172]+24*A[76]+128*A[116]+160*A[140], 24*A[77]+42*A[173]+128*A[ 117]+160*A[141]], [106*A[174]+160*A[142]+152*A[78]+128*A[118], 234*A[175]+152*A [79]+128*A[119]+160*A[143]], [192*A[96]+32*A[144]+136*A[80]+114*A[176]+128*A[64 ], 242*A[177]+128*A[65]+192*A[97]+136*A[81]+32*A[145]], [192*A[98]+32*A[146]+50 *A[178]+128*A[66]+8*A[82], 128*A[67]+178*A[179]+192*A[99]+32*A[147]+8*A[83]], [ 210*A[180]+200*A[84]+128*A[68]+32*A[148]+192*A[100], 128*A[69]+200*A[85]+192*A[ 101]+82*A[181]+32*A[149]], [32*A[150]+192*A[102]+128*A[70]+146*A[182]+72*A[86], 18*A[183]+32*A[151]+192*A[103]+128*A[71]+72*A[87]], [34*A[184]+168*A[88]+32*A[ 152]+128*A[72]+64*A[104], 162*A[185]+32*A[153]+64*A[105]+168*A[89]+128*A[73]], [128*A[74]+40*A[90]+32*A[154]+226*A[186]+64*A[106], 32*A[155]+98*A[187]+128*A[ 75]+64*A[107]+40*A[91]], [130*A[188]+32*A[156]+64*A[108]+128*A[76]+232*A[92], 128*A[77]+32*A[157]+2*A[189]+64*A[109]+232*A[93]], [104*A[94]+128*A[78]+66*A[ 190]+32*A[158]+64*A[110], 128*A[79]+32*A[159]+194*A[191]+64*A[111]+104*A[95]]], [1, 1, 1, 1, 2, 1, 5, 1, 14, 2, 132, 1, 42, 5, 173, 1, 150, 14, 140, 2, 156, 132, 8, 1, 254, 42, 244, 5, 162, 173, 125, 1, 230, 150, 124, 14, 100, 140, 216, 2, 44, 156, 24, 132, 56, 8, 16, 1, 246, 254, 172, 42, 140, 244, 104, 5, 214, 162, 164, 173, 210, 125, 29, 1, 134, 230, 220, 150, 20, 124, 56, 14, 116, 100, 104, 140, 40, 216, 176, 2, 204, 44, 248, 156, 200, 24, 176, 132, 88, 56, 48, 8, 112, 16, 32, 1, 166, 246, 188, 254, 4, 172, 24, 42, 28, 140, 120, 244, 88, 104, 208, 5, 142, 214, 156, 162, 156, 164, 200, 173, 70, 210, 132, 125, 50, 29, 93, 1, 198, 134, 156, 230, 244, 220, 248, 150, 164, 20, 72, 124, 72, 56, 112, 14, 20, 116, 72, 100, 56, 104, 80, 140, 8, 40, 144, 216, 208, 176, 96, 2, 12, 204, 184, 44, 40, 248, 112, 156, 232, 200, 208, 24, 80, 176, 96, 132, 152, 88, 240, 56, 144, 48, 96, 8, 176, 112, 96, 16, 224, 32, 64, 6, 166, 92, 246, 84, 188, 184, 254, 20, 4, 168, 172, 232, 24, 48, 42, 188, 28, 88, 140, 232, 120, 112, 244, 248, 88, 112, 104, 176, 208, 160, 5, 190, 142, 44, 214, 84, 156, 248, 162, 108, 156, 24, 164, 248, 200, 144, 173, 126, 70, 252, 210, 60, 132, 136, 125, 38 , 50, 68, 29, 242, 93, 221, 70, 198, 28, 134, 180, 156, 120, 230, 132, 244, 8, 220, 136, 248, 240, 150, 68, 164, 40, 20, 216, 72, 16, 124, 168, 72, 208, 56, 16, 112, 224, 14, 84, 20, 8, 116, 152, 72, 16, 100, 24, 56, 48, 104, 176, 80, 160, 140, 200, 8, 208, 40, 176, 144, 32, 216, 144, 208, 32, 176, 160, 96, 192]] Just for fun, , C(100000000000000000000), modulo , 256, is , 0 and the next, 10, terms in the sequence C(n) modulo, 256, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ------------------------------------------------ This ends this thrilling article, that took, 251.996, seconds. to produce . 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