All the Linear Schemes whose size is less than, 500, for determining the, CentralDellanoy, numbers modulo prime and prime powers for the first, 4, primes and powers <=, 3 By Shalosh B. Ekhad Recall that the, CentralDellanoy, numbers, let's call them C(n), may defined as the constant term of n (1/x + 3 + 2 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, 500, states. for the first, 3, primes and as high powers as we can afford with our limit of, 500, 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 an automata with 1, states Here it is: [[[A[1], A[1]]], [1]] Just for fun, , C(100000000000000000000), modulo , 2, is , 1 and the next, 10, terms in the sequence C(n) modulo, 2, are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 -------------------------------------- For C(n) modulo, 4, there is an automata with 5, states Here it is: [[[A[2], A[3]], [A[2], A[4]], [3*A[2], A[5]], [A[2], 2*A[2]+2*A[3]+A[4]], [A[2] +2*A[3]+A[4]+A[5], A[5]]], [1, 1, 3, 1, 3]] Just for fun, , C(100000000000000000000), modulo , 4, is , 1 and the next, 10, terms in the sequence C(n) modulo, 4, are 3, 1, 3, 1, 3, 1, 3, 1, 3, 1 -------------------------------------- For C(n) modulo, 8, there is an automata with 15, states Here it is: [[[A[2], A[3]], [A[4], A[5]], [A[6], A[7]], [A[4], A[8]], [A[9], A[10]], [3*A[4 ], A[11]], [A[12], A[13]], [A[14], A[15]], [5*A[4], 4*A[6]+A[8]], [2*A[4]+2*A[6 ]+A[9], 4*A[4]+5*A[5]+5*A[6]+3*A[7]+3*A[8]+A[9]+4*A[10]], [4*A[4]+2*A[5]+2*A[6] +A[8]+A[9]+3*A[11], 4*A[4]+5*A[5]+A[6]+3*A[7]+2*A[8]+A[9]+4*A[10]+A[11]], [A[4] +4*A[6]+A[9]+3*A[12], 4*A[5]+2*A[8]+3*A[11]], [6*A[5]+6*A[6]+4*A[7]+3*A[8]+2*A[ 9]+4*A[10]+A[11]+A[12], 3*A[5]+7*A[6]+5*A[7]+2*A[8]+5*A[9]+4*A[10]+A[11]+2*A[12 ]], [A[4], 4*A[5]+2*A[6]+A[8]+A[9]+A[14]], [4*A[6]+A[9]+A[12]+A[14], 4*A[5]+2*A [7]+2*A[8]+A[10]]], [1, 1, 3, 1, 5, 3, 7, 1, 5, 5, 3, 7, 7, 1, 1]] Just for fun, , C(100000000000000000000), modulo , 8, is , 1 and the next, 10, terms in the sequence C(n) modulo, 8, are 3, 5, 7, 1, 3, 5, 7, 1, 3, 5 -------------------------------------- For C(n) modulo, 3, there is an automata with 1, states Here it is: [[[A[1], 0, A[1]]], [1]] Just for fun, , C(100000000000000000000), modulo , 3, is , 0 and the next, 10, terms in the sequence C(n) modulo, 3, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 9, there is an automata with 9, states Here it is: [[[A[2], A[3], A[4]], [A[2], A[5], A[6]], [3*A[2], 6*A[2]+3*A[3], A[7]], [4*A[2 ], A[5], A[9]], [0, 6*A[2], 3*A[3]], [7*A[2], 3*A[3]+A[5], A[6]], [3*A[2], 6*A[ 4]+A[7], 6*A[3]+A[7]], [0, 6*A[2], 6*A[4]+A[7]], [4*A[2], 3*A[3]+3*A[4]+A[5]+2* A[7], 6*A[3]+3*A[4]+A[6]+A[7]]], [1, 1, 3, 4, 0, 7, 3, 0, 4]] Just for fun, , C(100000000000000000000), modulo , 9, is , 0 and the next, 10, terms in the sequence C(n) modulo, 9, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 27, there is an automata with 43, 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[5], A[14], A[15]], [A[16], A[17], A[18]], [A[19], A[20], A[21]], [3*A [5], A[22], A[23]], [A[24], A[25], A[26]], [A[27], A[28], A[29]], [13*A[5], A[ 30], A[31]], [A[32], A[33], A[18]], [A[35], A[36], A[37]], [A[38], A[39], A[40] ], [A[41], A[42], A[43]], [9*A[5], 9*A[6]+18*A[11], 9*A[7]+9*A[12]], [6*A[5]+9* A[8], 18*A[6]+9*A[9]+6*A[14], 15*A[7]+9*A[10]], [18*A[8]+9*A[11]+3*A[16], 9*A[9 ]+9*A[12]+3*A[17], 9*A[13]+3*A[18]], [25*A[5], 12*A[6]+4*A[14], 7*A[15]], [15*A [8]+7*A[16], 6*A[9]+7*A[17], 24*A[10]+7*A[18]], [18*A[5]+15*A[11]+4*A[19], 18*A [6]+6*A[12]+4*A[20], 18*A[7]+24*A[13]+4*A[21]], [18*A[5]+3*A[16], 3*A[17]+6*A[ 22], 18*A[7]+3*A[18]], [18*A[6]+9*A[8]+9*A[11]+6*A[14]+3*A[19]+A[22], 3*A[7]+9* A[9]+9*A[12]+3*A[20]+8*A[23], 9*A[10]+3*A[21]], [24*A[5], 18*A[6]+6*A[14], 6*A[ 7]+6*A[23]], [15*A[5]+8*A[24], 18*A[6]+6*A[14]+3*A[22]+8*A[25], 9*A[10]+6*A[18] +6*A[23]], [9*A[5]+15*A[6]+18*A[11]+6*A[14]+6*A[19]+2*A[22]+3*A[24], 3*A[7]+18* A[12]+3*A[18]+6*A[20]+3*A[22]+4*A[23]+3*A[25]+2*A[26], 6*A[7]+3*A[18]+6*A[21]+6 *A[23]+5*A[26]], [9*A[5]+18*A[6]+6*A[14]+6*A[19]+A[22]+8*A[24]+4*A[27], 9*A[6]+ 3*A[14], 3*A[7]+3*A[23]], [9*A[5]+18*A[6]+6*A[14]+6*A[19]+A[22]+6*A[24]+4*A[27] , 9*A[6]+3*A[14]+3*A[22]+7*A[25], 6*A[7]+6*A[18]+9*A[23]+11*A[26]], [6*A[19]+4* A[24]+5*A[27]+6*A[14]+A[22]+18*A[6]+12*A[5], A[28]+6*A[14]+6*A[22]+8*A[25]+18*A [6], A[29]+6*A[18]+6*A[26]+6*A[23]], [2*A[24]+A[16]+6*A[8]+6*A[5], A[17]+4*A[14 ]+6*A[9]+2*A[25]+12*A[6]+2*A[30], 7*A[18]+6*A[26]+6*A[23]+6*A[10]], [7*A[19]+3* A[24]+6*A[27]+4*A[14]+A[22]+6*A[11]+12*A[6]+9*A[5]+2*A[30], A[20]+2*A[28]+2*A[ 14]+6*A[12]+A[25]+6*A[6]+A[30], 15*A[13]+7*A[21]], [4*A[32]+10*A[24]+2*A[16]+18 *A[8]+12*A[5], 4*A[14]+4*A[22]+12*A[6]+2*A[30], 4*A[23]+2*A[31]+A[15]+6*A[7]], [4*A[32]+4*A[24]+2*A[16]+18*A[8]+18*A[5], 4*A[14]+3*A[22]+3*A[25]+12*A[6]+2*A[ 30], A[29]+2*A[26]+9*A[23]+A[31]+2*A[15]+6*A[7]+6*A[21]], [4*A[32]+5*A[24]+2*A[ 16]+5*A[14]+2*A[22]+18*A[8]+12*A[6]+15*A[5]+A[30], 7*A[23]+A[31]+2*A[15]+2*A[14 ]+A[25]+9*A[7]+6*A[6]+A[30], 2*A[18]+5*A[26]+9*A[23]+2*A[31]+4*A[34]+A[15]+9*A[ 13]+18*A[10]+6*A[7]], [4*A[32]+6*A[19]+2*A[24]+2*A[16]+5*A[27]+6*A[35]+5*A[14]+ 2*A[22]+18*A[8]+12*A[6]+A[5]+A[30], A[14]+3*A[22]+3*A[6], 9*A[23]+2*A[31]+2*A[ 15]+12*A[7]], [5*A[32]+3*A[19]+2*A[24]+2*A[16]+6*A[27]+6*A[35]+18*A[11]+15*A[8] +6*A[5], 2*A[17]+4*A[14]+5*A[33]+15*A[9]+2*A[25]+12*A[6]+2*A[30], 4*A[26]+9*A[ 23]+A[31]+A[34]+2*A[15]+6*A[10]+6*A[7]], [2*A[32]+3*A[19]+A[16]+2*A[27]+4*A[35] +4*A[14]+A[22]+15*A[11]+9*A[8]+12*A[6]+2*A[30], A[36]+4*A[28]+7*A[23]+A[31]+2*A [15]+4*A[14]+6*A[12]+2*A[25]+9*A[7]+12*A[6]+2*A[30], 9*A[23]+A[31]+5*A[37]+2*A[ 15]+15*A[13]+12*A[7]+2*A[21]], [0, 4*A[14]+A[22]+12*A[6]+2*A[30], 7*A[23]+A[31] +2*A[15]+9*A[7]], [7*A[19]+3*A[24]+6*A[27]+8*A[35]+5*A[14]+2*A[22]+9*A[11]+12*A [6]+6*A[5]+A[30], 4*A[14]+12*A[6]+2*A[30], 6*A[23]+2*A[31]+A[15]+6*A[7]], [4*A[ 32]+6*A[19]+2*A[24]+A[16]+A[38]+5*A[27]+6*A[35]+5*A[14]+2*A[22]+12*A[8]+12*A[6] +6*A[5]+A[30], 2*A[14]+A[25]+6*A[6]+A[30], A[40]+A[18]+A[26]+6*A[23]+A[31]+4*A[ 34]+2*A[15]+12*A[10]+6*A[7]], [8*A[32]+A[19]+2*A[24]+2*A[16]+6*A[41]+2*A[38]+4* A[27]+5*A[35]+24*A[8]+4*A[5], 2*A[36]+A[20]+2*A[40]+2*A[29]+2*A[18]+4*A[28]+4*A [26]+4*A[23]+4*A[37]+8*A[34]+4*A[14]+9*A[13]+9*A[12]+24*A[10]+2*A[25]+9*A[6]+2* A[21], A[40]+A[29]+A[18]+2*A[26]+6*A[23]+2*A[37]+4*A[34]+A[15]+18*A[13]+12*A[10 ]+3*A[7]+A[21]], [5*A[32]+2*A[16]+15*A[8], 8*A[36]+2*A[20]+8*A[28]+2*A[42]+5*A[ 14]+24*A[12]+3*A[33]+9*A[9]+7*A[25]+A[39]+12*A[6]+A[30], A[40]+3*A[34]+9*A[10]] , [8*A[32]+2*A[24]+2*A[16]+6*A[41]+2*A[38]+4*A[27]+7*A[35]+6*A[11]+24*A[8]+6*A[ 5], 7*A[36]+A[20]+6*A[28]+2*A[42]+3*A[14]+21*A[12]+3*A[25]+6*A[6], 2*A[40]+2*A[ 29]+2*A[18]+4*A[26]+9*A[23]+A[31]+A[37]+8*A[34]+2*A[15]+15*A[13]+24*A[10]+6*A[7 ]]], [1, 1, 3, 13, 1, 9, 25, 3, 24, 12, 13, 9, 22, 0, 7, 9, 6, 9, 25, 0, 16, 18 , 21, 24, 18, 15, 12, 18, 12, 0, 19, 9, 6, 9, 22, 0, 22, 0, 6, 0, 7, 0, 25]] Just for fun, , C(100000000000000000000), modulo , 27, is , 0 and the next, 10, terms in the sequence C(n) modulo, 27, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 5, there is an automata with 1, states Here it is: [[[A[1], 3*A[1], 3*A[1], 3*A[1], A[1]]], [1]] Just for fun, , C(100000000000000000000), modulo , 5, is , 2 and the next, 10, terms in the sequence C(n) modulo, 5, are 1, 1, 1, 2, 1, 3, 3, 3, 1, 1 -------------------------------------- For C(n) modulo, 25, there is an automata with 32, states Here it is: [[[A[2], A[3], A[4], A[5], A[6]], [A[2], A[7], A[8], A[9], A[10]], [3*A[2], A[ 11], A[12], A[13], A[14]], [13*A[2], A[15], A[16], A[17], A[18]], [13*A[2], A[ 19], A[12], A[21], A[22]], [21*A[2], A[23], A[8], A[25], A[26]], [8*A[2], 20*A[ 2]+20*A[3]+3*A[7], A[27], A[28], A[29]], [3*A[2], 15*A[2]+15*A[3]+3*A[7], 5*A[3 ]+5*A[4]+3*A[8], A[30], A[31]], [23*A[2], 20*A[2]+5*A[3]+3*A[7], 15*A[3]+15*A[4 ]+3*A[8], 10*A[4]+3*A[9], A[14]], [16*A[2], 5*A[3]+A[7], 20*A[4]+A[8], 10*A[5]+ A[9], A[10]], [14*A[2], 20*A[2]+20*A[3]+3*A[7]+2*A[11], 20*A[2]+20*A[3]+3*A[7]+ 4*A[8]+4*A[11], 5*A[4]+20*A[5]+4*A[9], 15*A[5]+15*A[6]+4*A[10]], [19*A[2], 10*A [2]+10*A[3]+3*A[11], 20*A[2]+20*A[3]+10*A[4]+3*A[7]+4*A[11]+3*A[12], 5*A[2]+10* A[3]+10*A[4]+2*A[7]+A[8]+4*A[9]+A[11]+3*A[12], 5*A[5]+20*A[6]+4*A[10]], [9*A[2] , 10*A[3]+3*A[11], 5*A[2]+10*A[3]+10*A[4]+2*A[7]+A[8]+A[11]+A[12], 15*A[2]+10*A [3]+20*A[4]+20*A[5]+A[7]+3*A[8]+4*A[9]+3*A[11]+4*A[12], 5*A[2]+10*A[3]+10*A[4]+ 20*A[5]+2*A[7]+A[8]+3*A[9]+4*A[10]+A[11]+3*A[12]+4*A[13]], [3*A[2]+10*A[3]+10*A [4]+20*A[5]+20*A[6]+2*A[7]+A[8]+3*A[9]+4*A[10]+A[11]+3*A[12]+4*A[13]+2*A[14], 15*A[3]+3*A[7], 15*A[4]+20*A[5]+10*A[6]+3*A[8]+4*A[9]+2*A[10]+2*A[13]+A[14], 15 *A[5]+10*A[6]+3*A[9]+A[10]+3*A[14], 3*A[10]], [4*A[2]+10*A[3]+20*A[4]+20*A[5]+ 10*A[6]+A[7]+3*A[8]+4*A[9]+2*A[10]+3*A[11]+4*A[12]+2*A[13]+A[14], 10*A[3]+10*A[ 4]+20*A[5]+20*A[6]+A[8]+3*A[9]+4*A[10]+3*A[11]+3*A[12]+4*A[13]+2*A[14], 10*A[4] +20*A[5]+10*A[6]+4*A[9]+2*A[10]+3*A[12]+2*A[13]+A[14], 10*A[5]+20*A[6]+3*A[10]+ 3*A[13]+4*A[14], 5*A[6]+3*A[14]], [4*A[2]+20*A[3]+20*A[4]+4*A[7]+3*A[8]+2*A[11] +2*A[12]+2*A[16], 10*A[3]+20*A[4]+A[7]+2*A[8]+A[11]+3*A[12]+3*A[16], 5*A[4]+4*A [8]+4*A[12]+A[16], 10*A[4]+10*A[5]+A[8]+A[12]+3*A[13]+2*A[16], 5*A[4]+10*A[5]+ 20*A[6]+4*A[8]+A[9]+4*A[10]+4*A[12]+3*A[13]+3*A[16]], [4*A[2]+20*A[3]+20*A[4]+4 *A[7]+3*A[8]+2*A[11]+2*A[12]+2*A[16], 10*A[3]+20*A[4]+2*A[8]+3*A[11]+3*A[12]+3* A[16], 5*A[4]+4*A[8]+2*A[12]+3*A[16], 5*A[4]+10*A[5]+4*A[8]+4*A[12]+3*A[16]+3*A [17], 20*A[4]+20*A[5]+10*A[6]+2*A[8]+4*A[9]+4*A[10]+2*A[12]+4*A[16]+2*A[17]], [ 3*A[2]+10*A[3]+20*A[4]+20*A[5]+2*A[7]+3*A[8]+3*A[9]+4*A[10]+A[11]+4*A[16]+4*A[ 17]+2*A[18], 15*A[3]+3*A[7], 15*A[4]+10*A[5]+5*A[6]+3*A[8]+2*A[9]+A[10]+A[17]+3 *A[18], 15*A[5]+5*A[6]+3*A[9]+3*A[10]+4*A[18], 3*A[10]], [4*A[2], 10*A[3]+20*A[ 4]+20*A[5]+3*A[8]+3*A[9]+4*A[10]+4*A[16]+4*A[17]+2*A[18]+3*A[19], 20*A[4]+2*A[8 ]+4*A[16], 20*A[5]+2*A[9]+2*A[10]+4*A[17]+A[18], 15*A[6]+2*A[10]+4*A[18]], [4*A [2]+20*A[3]+20*A[4]+4*A[7]+A[8]+4*A[16]+2*A[19]+4*A[20], 10*A[3]+20*A[4]+A[7]+A [8]+4*A[16]+A[19]+4*A[20], 5*A[4]+4*A[8]+A[16]+4*A[20], 10*A[4]+20*A[5]+A[8]+2* A[9]+2*A[16]+4*A[17]+A[20], 5*A[6]+3*A[18]], [4*A[2]+20*A[3]+20*A[4]+4*A[7]+A[8 ]+4*A[16]+2*A[19]+4*A[20], 10*A[3]+20*A[4]+4*A[8]+A[16]+3*A[19]+A[20], 5*A[4]+4 *A[8]+3*A[16]+2*A[20], 5*A[4]+10*A[5]+4*A[8]+3*A[16]+3*A[17]+4*A[20], 20*A[4]+ 10*A[5]+20*A[6]+2*A[8]+4*A[16]+4*A[17]+3*A[18]+2*A[20]+A[21]], [3*A[2]+10*A[3]+ 10*A[4]+10*A[5]+5*A[6]+A[7]+3*A[16]+2*A[17]+4*A[18]+3*A[19]+2*A[20]+3*A[21]+A[ 22], 15*A[3]+10*A[4]+10*A[5]+3*A[7]+A[16]+4*A[17]+3*A[18]+4*A[20]+A[21]+2*A[22] , 15*A[4]+4*A[16]+2*A[20], 15*A[5]+4*A[17]+3*A[18]+2*A[21]+2*A[22], 20*A[6]+4*A [18]+2*A[22]], [3*A[2]+10*A[3]+10*A[4]+10*A[5]+5*A[6]+A[7]+3*A[16]+2*A[17]+4*A[ 18]+3*A[19]+2*A[20]+3*A[21]+A[22], 5*A[3]+A[19], 5*A[4]+10*A[5]+15*A[6]+3*A[17] +A[18]+A[20]+2*A[21]+4*A[22], 5*A[5]+20*A[6]+2*A[18]+A[21]+3*A[22], 20*A[6]+A[ 22]], [3*A[2], 15*A[3]+10*A[4]+10*A[5]+15*A[6]+2*A[7]+2*A[16]+3*A[17]+A[18]+2*A [19]+3*A[20]+2*A[21]+4*A[22], 5*A[4]+10*A[5]+20*A[6]+A[16]+A[17]+2*A[18]+4*A[21 ]+3*A[22], 5*A[5]+20*A[6]+A[17]+2*A[18]+3*A[22], 15*A[6]+A[18]], [3*A[2]+20*A[3 ]+10*A[4]+10*A[5]+15*A[6]+4*A[7]+2*A[16]+3*A[17]+A[18]+2*A[19]+3*A[20]+2*A[21]+ 4*A[22], 5*A[3]+A[19], 5*A[4]+A[20], 5*A[5]+A[21], 5*A[6]+A[22]], [4*A[20]+A[19 ]+3*A[18]+4*A[17]+A[16]+2*A[22]+2*A[7]+A[2]+A[21]+10*A[3]+10*A[4]+10*A[5], 3*A[ 20]+A[18]+3*A[17]+2*A[16]+4*A[22]+A[7]+15*A[6]+2*A[21]+5*A[3]+10*A[4]+10*A[5], 4*A[20]+3*A[16]+15*A[4], A[18]+3*A[17]+4*A[22]+15*A[6]+4*A[21]+15*A[5], 3*A[18] +4*A[22]], [A[20]+4*A[19]+2*A[18]+A[17]+4*A[16]+3*A[22]+3*A[7]+20*A[6]+4*A[2]+4 *A[21]+20*A[3]+10*A[4]+10*A[5], 2*A[20]+A[19]+4*A[18]+2*A[17]+3*A[16]+A[22]+A[7 ]+5*A[6]+3*A[21]+10*A[3]+10*A[4]+10*A[5], 2*A[18]+A[17]+3*A[16]+3*A[22]+20*A[6] +4*A[21]+10*A[4]+10*A[5], 2*A[18]+3*A[17]+3*A[22]+20*A[6]+10*A[5], 3*A[18]+20*A [6]], [4*A[20]+A[19]+3*A[18]+4*A[17]+A[16]+2*A[22]+2*A[7]+4*A[2]+A[21]+10*A[3]+ 10*A[4]+10*A[5], 3*A[19]+10*A[3], 3*A[20]+10*A[4], 3*A[21]+10*A[5], 3*A[22]+10* A[6]], [3*A[2], 3*A[20]+A[18]+3*A[17]+2*A[16]+4*A[22]+3*A[7]+15*A[6]+2*A[21]+15 *A[3]+10*A[4]+10*A[5], 2*A[20]+3*A[18]+4*A[17]+4*A[16]+2*A[22]+A[21]+15*A[4]+10 *A[5], A[18]+4*A[17]+4*A[22]+15*A[6]+2*A[21]+15*A[5], 4*A[18]+2*A[22]+10*A[6]], [2*A[20]+3*A[19]+4*A[18]+2*A[17]+3*A[16]+A[22]+A[7]+5*A[6]+4*A[2]+3*A[21]+10*A[ 3]+10*A[4]+10*A[5], 4*A[20]+3*A[19]+3*A[18]+4*A[17]+A[16]+2*A[22]+A[21]+10*A[3] +10*A[4]+10*A[5], 3*A[20]+3*A[18]+4*A[17]+2*A[22]+A[21]+10*A[4]+10*A[5], 3*A[18 ]+2*A[22]+3*A[21]+10*A[5], 3*A[22]], [3*A[2]+10*A[3]+10*A[4]+10*A[5]+5*A[6]+A[7 ]+3*A[16]+2*A[17]+4*A[18]+3*A[19]+2*A[20]+3*A[21]+A[22], 15*A[3]+10*A[4]+10*A[5 ]+3*A[7]+A[16]+4*A[17]+3*A[18]+4*A[20]+A[21]+2*A[22], 15*A[4]+4*A[16]+2*A[20], 15*A[5]+4*A[17]+3*A[18]+2*A[21]+2*A[22], 20*A[6]+4*A[18]+2*A[22]], [4*A[20]+A[ 19]+3*A[18]+4*A[17]+A[16]+2*A[22]+2*A[7]+3*A[2]+A[21]+10*A[3]+10*A[4]+10*A[5], 15*A[3]+3*A[7], 2*A[20]+2*A[18]+A[17]+4*A[16]+3*A[22]+20*A[6]+4*A[21]+15*A[4]+ 10*A[5], 4*A[17]+2*A[21]+15*A[5], 4*A[18]+2*A[22]+5*A[6]]], [1, 1, 3, 13, 13, 21, 8, 3, 23, 16, 14, 19, 9, 23, 14, 19, 19, 23, 4, 19, 19, 13, 13, 3, 18, 21, 9, 24, 3, 14, 13, 23]] Just for fun, , C(100000000000000000000), modulo , 25, is , 22 and the next, 10, terms in the sequence C(n) modulo, 25, are 16, 11, 11, 12, 1, 8, 8, 13, 11, 16 -------------------------------------- For C(n) modulo, 125, there is an automata with 207, 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[7], 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]], [3*A[7], 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]], [13*A[7], 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]], [63*A[7], 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]], [71*A[7], A[128], A[129], A[130], A [131]], [A[132], A[133], A[134], A[135], A[136]], [A[137], A[138], A[43], 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]], [58*A[7], 100*A[27]+28*A[32]+105*A[8], 75*A[28]+55* A[9]+28*A[33], 5*A[10]+28*A[34]+50*A[29], 105*A[11]+3*A[35]+25*A[30]], [13*A[36 ]+85*A[12]+45*A[7], 105*A[8]+35*A[13]+40*A[32]+13*A[37], 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]], [78*A[7], 8*A[32]+20*A[8]+75* A[27], 25*A[28]+8*A[33]+95*A[9], 100*A[29]+8*A[34]+45*A[10], 120*A[11]+8*A[35]+ 50*A[30]], [10*A[12]+8*A[36]+90*A[7], 85*A[13]+15*A[32]+8*A[37]+25*A[8], 35*A[ 14]+15*A[33]+8*A[38]+100*A[9], 8*A[39]+50*A[10]+110*A[15]+15*A[34], 15*A[35]+8* A[40]+60*A[16]], [5*A[36]+35*A[17]+23*A[41]+100*A[12]+25*A[7], 5*A[37]+23*A[42] +25*A[8]+50*A[13]+110*A[18], 23*A[43]+25*A[9]+60*A[19]+5*A[38], A[190], A[191]] , [A[192], A[193], A[194], A[195], A[196]], [A[197], A[198], A[199], A[200], A[ 201]], [73*A[7], 8*A[32]+100*A[27]+115*A[8], 75*A[28]+8*A[33]+40*A[9], 50*A[29] +8*A[34]+90*A[10], 8*A[35]+15*A[11]+25*A[30]], [23*A[36]+10*A[12]+95*A[7], 20*A [32]+23*A[37]+60*A[13]+50*A[8], 23*A[38]+110*A[14]+20*A[33]+25*A[9], 20*A[34]+ 35*A[15]+23*A[39], 23*A[40]+85*A[16]+20*A[35]+100*A[11]], [15*A[36]+10*A[17]+8* A[41]+75*A[12], 60*A[18]+15*A[37]+8*A[42]+50*A[13], 110*A[19]+15*A[38]+8*A[43]+ 25*A[14], 35*A[20]+8*A[44]+15*A[39], 15*A[40]+100*A[16]+8*A[45]+85*A[21]], [18* A[46]+100*A[17]+10*A[41]+10*A[22], 75*A[18]+60*A[23]+10*A[42]+18*A[47], 50*A[19 ]+18*A[48]+110*A[24]+10*A[43], 25*A[20]+10*A[44]+18*A[49]+35*A[25], A[202]], [A [203], A[204], A[205], A[206], A[207]], [66*A[7], 11*A[32]+5*A[8], 11*A[33]+80* A[9], 11*A[34]+30*A[10], 11*A[35]+105*A[11]], [11*A[36]+45*A[12]+50*A[7], 11*A[ 37]+120*A[13]+50*A[8], 11*A[38]+70*A[14]+50*A[9], 20*A[15]+50*A[10]+11*A[39], 11*A[40]+95*A[16]+50*A[11]], [70*A[17]+A[41], 20*A[18]+A[42], 95*A[19]+A[43], 45*A[20]+A[44], A[45]+120*A[21]], [16*A[46]+100*A[17]+95*A[22], 100*A[18]+45*A[ 23]+16*A[47], 100*A[19]+16*A[48]+120*A[24], 100*A[20]+16*A[49]+70*A[25], 16*A[ 50]+20*A[26]+100*A[21]], [45*A[27]+100*A[22]+6*A[51], 6*A[52]+120*A[28]+100*A[ 23], 6*A[53]+70*A[29]+100*A[24], 6*A[54]+100*A[25]+20*A[30], 6*A[55]+100*A[26]+ 95*A[31]], [18*A[36]+60*A[12]+100*A[7], 10*A[32]+18*A[37]+15*A[56]+60*A[13]+45* A[8], 18*A[38]+60*A[14]+100*A[9], 60*A[15]+100*A[10]+18*A[39], 18*A[40]+60*A[16 ]+100*A[11]], [10*A[17]+18*A[41]+75*A[12], 10*A[18]+18*A[42]+75*A[13], 10*A[19] +18*A[43]+75*A[14], 10*A[20]+18*A[44]+75*A[15], 75*A[16]+18*A[45]+10*A[21]], [ 18*A[46]+50*A[17]+85*A[22], 50*A[18]+85*A[23]+18*A[47], 50*A[19]+18*A[48]+85*A[ 24], 50*A[20]+18*A[49]+85*A[25], 18*A[50]+85*A[26]+50*A[21]], [14*A[32]+7*A[56] +85*A[27]+25*A[22]+65*A[8]+18*A[51], 11*A[57]+18*A[52]+85*A[28]+25*A[23]+12*A[ 33]+55*A[9], 18*A[53]+85*A[29]+25*A[24]+3*A[34]+19*A[58]+40*A[10], 48*A[59]+18* A[54]+A[35]+80*A[11]+25*A[25]+85*A[30], 18*A[55]+25*A[26]+35*A[31]], [44*A[36]+ 42*A[60]+75*A[12]+49*A[7], 30*A[32]+23*A[56]+15*A[8], 23*A[57]+30*A[33]+15*A[9] , 18*A[58]+60*A[10], 18*A[59]+60*A[11]], [25*A[36]+28*A[60]+5*A[12]+100*A[7], 40*A[32]+39*A[37]+20*A[56]+30*A[61]+55*A[13]+35*A[8], 15*A[57]+4*A[38]+105*A[14 ]+20*A[33]+95*A[9], 20*A[58]+80*A[15]+25*A[10]+4*A[39], 25*A[59]+4*A[40]+55*A[ 16]+60*A[11]], [28*A[36]+55*A[17]+44*A[60]+24*A[41]+50*A[12]+120*A[7], 50*A[32] +30*A[18]+41*A[37]+15*A[56]+24*A[42]+28*A[61]+25*A[13]+95*A[8], 15*A[57]+5*A[19 ]+44*A[38]+24*A[43]+32*A[62]+50*A[14]+50*A[33]+90*A[9], 105*A[20]+15*A[58]+24*A [44]+30*A[10]+20*A[39], 15*A[59]+20*A[40]+50*A[16]+30*A[11]+24*A[45]+80*A[21]], [36*A[36]+19*A[46]+50*A[17]+48*A[60]+5*A[41]+105*A[22]+75*A[12]+15*A[7], 40*A[ 32]+100*A[18]+80*A[23]+12*A[37]+20*A[56]+5*A[42]+16*A[61]+100*A[13]+19*A[47]+55 *A[8], 10*A[57]+25*A[19]+19*A[48]+55*A[24]+9*A[38]+5*A[43]+12*A[62]+75*A[14]+20 *A[33]+85*A[9], 75*A[20]+30*A[58]+31*A[63]+5*A[44]+25*A[15]+19*A[49]+65*A[10]+ 30*A[25]+42*A[39], 19*A[50]+5*A[26]+100*A[16]+5*A[45]], [14*A[32]+15*A[46]+75*A [17]+7*A[56]+80*A[27]+100*A[22]+65*A[8]+14*A[51], 19*A[57]+14*A[52]+75*A[18]+80 *A[28]+25*A[23]+38*A[33]+15*A[47]+55*A[9], 14*A[53]+75*A[19]+15*A[48]+80*A[29]+ 75*A[24]+A[34]+28*A[58]+16*A[63]+100*A[15]+70*A[10]+12*A[39], 75*A[20]+14*A[54] +15*A[49]+80*A[30], 15*A[50]+14*A[55]+50*A[26]+55*A[31]+75*A[21]], [33*A[36]+44 *A[60]+50*A[12]+14*A[7], 31*A[32]+21*A[56]+25*A[8], 6*A[57]+21*A[33]+105*A[9], A[34]+11*A[58]+4*A[63]+25*A[15]+30*A[10]+3*A[39], 11*A[59]+A[35]+35*A[11]], [45 *A[36]+48*A[60]+80*A[12]+35*A[7], 30*A[32]+100*A[18]+56*A[37]+36*A[42]+31*A[61] +105*A[13]+120*A[8]+8*A[66], 20*A[57]+17*A[38]+19*A[62]+105*A[14]+30*A[33], 40* A[58]+52*A[63]+105*A[15]+85*A[10]+48*A[39], 25*A[59]+8*A[40]+30*A[16]+7*A[64]+ 55*A[11]], [19*A[36]+55*A[17]+12*A[60]+7*A[65]+23*A[41]+100*A[12]+60*A[7], 20*A [32]+5*A[18]+19*A[37]+23*A[42]+12*A[61]+100*A[13]+90*A[8]+7*A[66], 5*A[57]+9*A[ 67]+80*A[19]+51*A[38]+32*A[43]+28*A[62]+75*A[14]+30*A[33]+120*A[9], 105*A[20]+ 35*A[58]+23*A[63]+4*A[44]+85*A[10]+26*A[39], 30*A[59]+26*A[40]+23*A[64]+50*A[11 ]+4*A[45]+80*A[21]], [45*A[36]+24*A[46]+50*A[17]+30*A[60]+6*A[65]+12*A[41]+30*A [22]+50*A[12], 15*A[32]+5*A[23]+30*A[37]+30*A[42]+15*A[61]+25*A[13]+24*A[47]+60 *A[8]+10*A[66], 10*A[67]+100*A[19]+24*A[48]+105*A[24]+30*A[38]+30*A[43]+15*A[62 ]+25*A[14]+15*A[33]+60*A[9], 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[50*A[36]+12*A[46]+100*A[17]+38*A[60]+15*A[65]+20*A[41]+6*A[70]+75*A[ 22]+80*A[12]+50*A[7], 40*A[32]+25*A[18]+52*A[37]+9*A[42]+39*A[61]+105*A[13]+40* A[8]+2*A[66], 15*A[57]+11*A[38]+11*A[62]+55*A[14]+30*A[33]+15*A[9], 40*A[58]+45 *A[63]+80*A[15]+90*A[10]+49*A[39], 20*A[59]+20*A[40]+5*A[16]+23*A[64]+40*A[11]] , [19*A[36]+4*A[46]+80*A[17]+12*A[60]+9*A[65]+22*A[41]+2*A[70]+25*A[22]+100*A[ 12]+60*A[7], 40*A[32]+30*A[18]+50*A[37]+29*A[42]+16*A[71]+40*A[61]+75*A[13]+32* A[47]+35*A[8]+25*A[66], 34*A[67]+80*A[19]+52*A[48]+50*A[24]+26*A[72]+46*A[38]+ 32*A[43]+48*A[62]+100*A[14]+60*A[33]+5*A[9], 80*A[20]+35*A[58]+41*A[63]+16*A[44 ]+75*A[10]+42*A[39]+6*A[68], 50*A[59]+20*A[40]+100*A[16]+13*A[69]+10*A[64]+75*A [11]+10*A[45]+80*A[21]], [38*A[36]+29*A[46]+75*A[17]+44*A[60]+28*A[65]+56*A[41] +30*A[70]+5*A[22]+95*A[7], 55*A[32]+30*A[23]+44*A[37]+63*A[42]+32*A[71]+47*A[61 ]+25*A[13]+33*A[47]+105*A[8]+34*A[66], 17*A[67]+25*A[19]+10*A[48]+55*A[24]+8*A[ 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A[197]+2*A[186]+A[158]+12*A[32]+16*A[142]+A[52]+60*A[18]+2*A[46] +120*A[17]+80*A[28]+35*A[23]+24*A[41]+3*A[193]+2*A[176]+8*A[148]+2*A[133]+15*A[ 143]+2*A[56]+4*A[203]+A[42]+80*A[27]+35*A[22]+13*A[167]+40*A[13]+10*A[168]+4*A[ 147]+22*A[162]+A[185]+8*A[153]+3*A[47]+55*A[8]+16*A[163]+2*A[137]+4*A[157]+12*A [138]+9*A[181], 3*A[194]+15*A[144]+28*A[154]+20*A[164]+2*A[187]+A[53]+60*A[19]+ 4*A[48]+80*A[29]+4*A[172]+55*A[24]+12*A[139]+20*A[134]+4*A[177]+A[43]+8*A[149]+ 75*A[14]+3*A[33]+10*A[169]+15*A[9]+9*A[182], 2*A[188]+3*A[195]+6*A[155]+9*A[183 ]+8*A[135]+60*A[20]+3*A[53]+60*A[29]+15*A[145]+8*A[150]+2*A[160]+A[54]+A[58]+A[ 44]+60*A[15]+4*A[165]+12*A[140]+2*A[173]+5*A[10]+12*A[169]+100*A[25]+10*A[170]+ 80*A[30], 3*A[202]+2*A[50]+3*A[196]+A[174]+6*A[136]+11*A[156]+85*A[26]+70*A[16] +3*A[131]+55*A[31]+4*A[189]+4*A[161]+21*A[146]+16*A[166]+15*A[11]+4*A[171]+2*A[ 179]+21*A[184]+100*A[21]+24*A[151]+20*A[141]], [4*A[197]+19*A[142]+4*A[192]+4*A [46]+55*A[17]+12*A[60]+36*A[41]+12*A[180]+12*A[132]+A[203]+50*A[27]+4*A[175]+10 *A[22]+10*A[167]+50*A[12]+34*A[162]+23*A[152]+3*A[137]+4*A[157]+4*A[84], A[197] +2*A[158]+9*A[32]+16*A[142]+15*A[18]+2*A[46]+120*A[17]+24*A[41]+2*A[176]+4*A[ 133]+2*A[56]+4*A[203]+4*A[42]+80*A[27]+35*A[22]+13*A[167]+A[61]+65*A[13]+4*A[ 147]+22*A[162]+A[185]+11*A[153]+40*A[8]+2*A[137]+4*A[157]+28*A[138], 17*A[154]+ 3*A[159]+110*A[19]+4*A[172]+22*A[139]+6*A[134]+3*A[177]+A[43]+A[62]+105*A[14], 2*A[195]+2*A[190]+9*A[155]+12*A[183]+4*A[135]+80*A[20]+10*A[145]+3*A[44]+55*A[ 15]+2*A[49]+16*A[165]+16*A[140]+15*A[25], A[202]+4*A[50]+4*A[196]+4*A[136]+9*A[ 156]+20*A[26]+55*A[16]+90*A[31]+2*A[189]+4*A[161]+2*A[191]+22*A[146]+28*A[166]+ 2*A[171]+22*A[184]+100*A[21]+12*A[151]+20*A[141]], [2*A[197]+17*A[142]+A[192]+2 *A[46]+115*A[17]+23*A[41]+5*A[180]+3*A[203]+50*A[27]+55*A[22]+10*A[167]+23*A[ 162]+4*A[137]+A[157], 2*A[204]+A[186]+A[158]+115*A[18]+80*A[28]+55*A[23]+A[193] +4*A[148]+18*A[143]+4*A[42]+13*A[168]+A[47]+23*A[163]+23*A[138]+3*A[198]+5*A[ 181], 4*A[205]+A[199]+A[194]+31*A[144]+33*A[164]+2*A[159]+4*A[187]+95*A[19]+3*A [48]+45*A[29]+30*A[24]+19*A[139]+2*A[43]+16*A[149]+22*A[169]+5*A[182], 3*A[188] +A[195]+5*A[183]+95*A[20]+15*A[145]+12*A[150]+2*A[160]+4*A[54]+2*A[44]+3*A[49]+ 19*A[165]+19*A[140]+35*A[25]+25*A[170]+45*A[30], 4*A[202]+2*A[196]+35*A[26]+10* A[31]+3*A[189]+3*A[161]+18*A[146]+7*A[166]+3*A[171]+24*A[184]+75*A[21]+18*A[151 ]+15*A[141]], [2*A[197]+12*A[142]+2*A[46]+120*A[17]+24*A[41]+3*A[203]+50*A[27]+ 30*A[22]+10*A[167]+24*A[162]+2*A[137]+4*A[157], 2*A[204]+A[186]+3*A[158]+15*A[ 18]+80*A[28]+70*A[23]+4*A[148]+13*A[143]+4*A[42]+13*A[168]+3*A[47]+32*A[163]+28 *A[138]+3*A[198], 4*A[205]+A[199]+26*A[144]+30*A[164]+4*A[187]+2*A[48]+45*A[29] +110*A[24]+16*A[149]+22*A[169], 2*A[188]+4*A[200]+80*A[20]+14*A[145]+8*A[150]+A [160]+4*A[54]+A[206]+3*A[44]+24*A[165]+16*A[140]+30*A[25]+32*A[170]+65*A[30], A [202]+A[50]+4*A[196]+60*A[26]+10*A[31]+3*A[189]+22*A[146]+12*A[166]+3*A[171]+16 *A[184]+18*A[151]], [4*A[197]+19*A[142]+3*A[46]+80*A[17]+16*A[41]+A[203]+60*A[ 27]+115*A[22]+23*A[167]+18*A[147]+38*A[162]+4*A[185]+3*A[137]+A[157], 3*A[204]+ 2*A[186]+3*A[158]+15*A[18]+55*A[23]+10*A[148]+17*A[143]+4*A[42]+17*A[168]+2*A[ 47]+26*A[163]+28*A[138]+2*A[198], 4*A[205]+A[199]+16*A[144]+14*A[164]+A[159]+A[ 187]+80*A[19]+95*A[29]+120*A[24]+16*A[139]+3*A[43]+6*A[149]+14*A[169], 2*A[150] +2*A[54]+9*A[170]+55*A[30], A[202]+A[207]+4*A[196]+65*A[26]+90*A[31]+4*A[189]+ 27*A[146]+10*A[166]+A[171]+16*A[184]+17*A[151]]], [1, 1, 3, 13, 63, 71, 1, 58, 78, 73, 66, 3, 114, 94, 109, 48, 13, 14, 94, 44, 48, 63, 104, 119, 119, 13, 71, 63, 3, 68, 46, 83, 28, 123, 41, 58, 54, 109, 24, 28, 78, 84, 114, 89, 88, 73, 84, 99, 24, 23, 66, 73, 113, 78, 16, 74, 9, 69, 123, 114, 7, 122, 92, 49, 94, 107, 122, 97, 74, 109, 47, 17, 117, 34, 48, 94, 64, 109, 23, 4, 14, 74, 108, 14 , 32, 22, 92, 99, 94, 82, 122, 97, 49, 44, 27, 97, 97, 19, 48, 69, 39, 9, 98, 104, 39, 99, 108, 104, 27, 67, 12, 89, 119, 82, 122, 122, 49, 119, 27, 72, 72, 119, 13, 39, 34, 104, 13, 118, 63, 33, 86, 63, 44, 74, 64, 8, 3, 34, 114, 14, 38, 68, 94, 9, 109, 43, 46, 113, 78, 93, 46, 83, 4, 59, 99, 53, 28, 59, 89, 64, 38, 123, 109, 124, 49, 73, 41, 123, 38, 3, 116, 117, 37, 89, 109, 27, 67, 42, 14, 24, 117, 37, 12, 74, 28, 34, 79, 74, 3, 107, 94, 89, 12, 57, 57, 64, 88, 64 , 9, 79, 13, 24, 23, 69, 89, 59, 23]] Just for fun, , C(100000000000000000000), modulo , 125, is , 22 and the next, 10, terms in the sequence C(n) modulo, 125, are 66, 36, 11, 62, 26, 8, 58, 38, 11, 91 -------------------------------------- For C(n) modulo, 7, there is an automata with 1, states Here it is: [[[A[1], 3*A[1], 6*A[1], 0, 6*A[1], 3*A[1], A[1]]], [1]] Just for fun, , C(100000000000000000000), modulo , 7, is , 0 and the next, 10, terms in the sequence C(n) modulo, 7, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 -------------------------------------- For C(n) modulo, 49, there is an automata with 59, states Here it is: [[[A[2], A[3], A[4], A[5], A[6], A[7], A[8]], [A[2], A[9], A[10], A[11], A[12], A[13], A[14]], [3*A[2], A[15], A[16], A[17], A[18], A[19], A[20]], [13*A[2], A[ 21], A[22], A[23], A[24], A[25], A[26]], [14*A[2], 35*A[2]+7*A[3], 21*A[3], 7*A [4]+42*A[5], A[27], A[28], A[29]], [27*A[2], A[30], A[31], A[23], A[33], A[34], A[35]], [17*A[2], A[36], A[37], A[17], A[39], A[40], A[41]], [22*A[2], A[42], A [43], A[11], A[45], A[46], A[47]], [31*A[2], 42*A[2]+3*A[9], A[48], A[49], A[39 ], A[51], A[52]], [41*A[2], 42*A[2]+21*A[3]+6*A[9], 35*A[3]+7*A[4]+6*A[10], A[ 53], A[33], A[55], A[56]], [42*A[2], 35*A[2]+35*A[3], 21*A[3]+28*A[4], 7*A[4]+ 21*A[5], 42*A[5]+14*A[6], 28*A[6]+7*A[7], 14*A[7]], [48*A[2], 42*A[2]+7*A[3]+6* A[9], 35*A[3]+21*A[4]+6*A[10], 28*A[4]+35*A[5]+6*A[11], 21*A[5]+6*A[12], A[55], A[58]], [38*A[2], 42*A[2]+35*A[3]+3*A[9], 35*A[3]+35*A[4]+3*A[10], 28*A[4]+35*A [5]+3*A[11], 21*A[5]+35*A[6]+3*A[12], 14*A[6]+35*A[7]+3*A[13], A[59]], [43*A[2] , 35*A[3]+A[9], 28*A[4]+A[10], 21*A[5]+A[11], 14*A[6]+A[12], 7*A[7]+A[13], A[14 ]], [2*A[2], 28*A[2]+14*A[3]+2*A[9], 7*A[2]+28*A[3]+28*A[4]+4*A[9]+2*A[10]+A[15 ], 35*A[4]+42*A[5]+2*A[11], 14*A[5]+7*A[6]+2*A[12], 42*A[6]+21*A[7]+2*A[13], 21 *A[7]+35*A[8]+2*A[14]], [39*A[2], 42*A[2]+42*A[3]+5*A[9]+2*A[15], 42*A[2]+42*A[ 3]+28*A[4]+3*A[9]+6*A[15]+6*A[16], 42*A[2]+42*A[3]+42*A[4]+7*A[5]+3*A[9]+5*A[10 ]+4*A[11]+6*A[15]+3*A[16], 14*A[5]+14*A[6]+4*A[12], 42*A[6]+21*A[7]+4*A[13], 21 *A[7]+28*A[8]+4*A[14]], [0, 42*A[3]+28*A[4]+42*A[5]+3*A[9]+4*A[10]+3*A[11]+6*A[ 15]+A[16]+6*A[17], 42*A[4]+42*A[5]+3*A[10]+5*A[11]+6*A[16]+3*A[17], 28*A[5]+3*A [11]+6*A[17], 28*A[5]+14*A[6], 35*A[6]+42*A[7], 42*A[7]+21*A[8]], [4*A[2]+42*A[ 3]+42*A[4]+28*A[5]+3*A[9]+5*A[10]+2*A[11]+6*A[15]+3*A[16]+4*A[17], 28*A[3]+2*A[ 9]+3*A[15], 28*A[4]+28*A[5]+2*A[10]+4*A[11]+3*A[16]+A[17], 35*A[5]+2*A[11]+3*A[ 17], 14*A[5]+28*A[6]+4*A[12], 7*A[5]+14*A[6]+21*A[7]+A[12]+4*A[13]+2*A[18], 21* A[7]+14*A[8]+4*A[14]], [2*A[2]+28*A[3]+14*A[4]+7*A[5]+2*A[9]+A[10]+6*A[11]+4*A[ 15]+2*A[16]+5*A[17], 14*A[3]+28*A[4]+42*A[5]+4*A[10]+3*A[11]+3*A[15]+A[16]+6*A[ 17], 14*A[4]+42*A[5]+3*A[11]+3*A[16]+6*A[17], 21*A[5]+3*A[17], 7*A[5]+28*A[6]+A [12]+5*A[18], 42*A[5]+7*A[6]+7*A[7]+6*A[12]+5*A[13]+5*A[18]+6*A[19], 7*A[5]+14* A[6]+28*A[7]+21*A[8]+A[12]+2*A[13]+2*A[14]+2*A[18]+4*A[19]], [3*A[2]+42*A[3]+7* A[4]+14*A[5]+5*A[9]+6*A[10]+A[11]+3*A[15]+5*A[16]+2*A[17], 21*A[3]+28*A[4]+42*A [5]+3*A[9]+4*A[10]+3*A[11]+A[16]+6*A[17], 21*A[4]+28*A[5]+3*A[10]+2*A[11]+4*A[ 17], 21*A[5]+14*A[6]+28*A[7]+21*A[8]+3*A[11]+A[12]+2*A[13]+2*A[14]+2*A[18]+4*A[ 19]+4*A[20], 21*A[6]+7*A[7]+35*A[8]+3*A[12]+6*A[13]+6*A[14]+5*A[19]+5*A[20], 21 *A[7]+3*A[13], 3*A[14]], [4*A[2]+42*A[3]+42*A[4]+28*A[5]+6*A[9]+5*A[10]+2*A[11] +3*A[16]+4*A[17]+6*A[21], 28*A[3]+42*A[4]+28*A[5]+3*A[9]+3*A[10]+4*A[11]+6*A[16 ]+A[17]+6*A[21], 28*A[4]+28*A[5]+2*A[11]+6*A[16]+4*A[17], 28*A[5]+42*A[6]+42*A[ 7]+7*A[8]+5*A[12]+3*A[13]+3*A[14]+6*A[17]+3*A[18]+6*A[19]+6*A[20], 28*A[6]+28*A [7]+14*A[8]+4*A[13]+4*A[14]+6*A[18]+A[19]+A[20], 28*A[7]+35*A[8]+A[14]+6*A[19]+ 2*A[20], 42*A[8]+6*A[20]], [A[2]+21*A[3]+14*A[4]+3*A[9]+A[10]+A[16]+3*A[21]+4*A [22], 7*A[3]+7*A[4]+5*A[10]+5*A[16]+6*A[21]+6*A[22], 5*A[10]+4*A[16]+2*A[22], 7 *A[5]+42*A[6]+42*A[7]+7*A[8]+6*A[10]+A[11]+5*A[12]+3*A[13]+3*A[14]+2*A[16]+3*A[ 18]+6*A[19]+6*A[20]+5*A[22], 4*A[12]+6*A[18], 35*A[8]+4*A[13]+6*A[14]+6*A[19]+5 *A[20], 21*A[8]+4*A[14]+6*A[20]], [21*A[3]+35*A[4]+21*A[5]+42*A[6]+42*A[7]+7*A[ 8]+3*A[9]+5*A[10]+3*A[11]+5*A[12]+3*A[13]+3*A[14]+3*A[18]+6*A[19]+6*A[20]+3*A[ 21]+5*A[22]+3*A[23], 35*A[3]+5*A[9]+5*A[21], 35*A[4]+7*A[5]+28*A[6]+14*A[7]+35* A[8]+5*A[10]+A[11]+4*A[12]+A[13]+A[14]+A[18]+2*A[19]+2*A[20]+5*A[22]+A[23], 35* A[5]+7*A[6]+42*A[7]+5*A[11]+6*A[12]+5*A[13]+5*A[14]+5*A[18]+3*A[19]+3*A[20]+5*A [23], 7*A[6]+6*A[12]+5*A[18], 7*A[7]+6*A[13]+5*A[19], 7*A[8]+6*A[14]+5*A[20]], [A[2]+7*A[3]+28*A[4]+7*A[5]+35*A[6]+A[9]+4*A[10]+A[11]+2*A[12]+5*A[18]+A[21]+4* A[22]+A[23]+3*A[24], 7*A[3]+35*A[4]+21*A[5]+42*A[6]+5*A[10]+3*A[11]+6*A[12]+A[ 18]+6*A[21]+5*A[22]+3*A[23]+2*A[24], 7*A[4]+6*A[22], 7*A[5]+35*A[6]+2*A[12]+5*A [18]+6*A[23]+3*A[24], 4*A[12]+6*A[18], 21*A[7]+6*A[12]+2*A[18]+5*A[19]+5*A[24], 42*A[7]+6*A[12]+3*A[13]+2*A[18]+6*A[19]+5*A[20]+5*A[24]], [4*A[2]+28*A[3]+14*A[ 4]+28*A[5]+35*A[6]+4*A[9]+2*A[10]+4*A[11]+A[12]+6*A[18]+4*A[21]+2*A[22]+4*A[23] +5*A[24], 28*A[3]+3*A[9]+6*A[21], 28*A[4]+21*A[5]+42*A[6]+3*A[10]+3*A[11]+6*A[ 12]+A[18]+6*A[22]+3*A[23]+2*A[24], 28*A[5]+35*A[6]+3*A[11]+2*A[12]+5*A[18]+6*A[ 23]+3*A[24], 6*A[12]+A[18]+5*A[24], 28*A[6]+14*A[7]+A[12]+5*A[18]+A[19]+2*A[24] +6*A[25], 35*A[8]+6*A[20]], [6*A[2], 2*A[20]+A[19]+3*A[18]+2*A[24]+6*A[26]+5*A[ 23]+6*A[22]+5*A[11]+6*A[10]+6*A[9]+3*A[25]+21*A[8]+7*A[7]+14*A[6]+42*A[3]+42*A[ 4]+35*A[5], A[20]+4*A[19]+5*A[18]+A[24]+3*A[26]+6*A[23]+6*A[11]+6*A[10]+5*A[25] +35*A[8]+21*A[7]+21*A[6]+42*A[4]+42*A[5], A[20]+4*A[19]+5*A[18]+A[24]+3*A[26]+6 *A[11]+5*A[25]+35*A[8]+21*A[7]+21*A[6]+42*A[5], 5*A[18]+2*A[24]+28*A[6], A[20]+ 5*A[19]+3*A[26]+2*A[25]+35*A[8]+28*A[7], 5*A[20]+2*A[26]+35*A[8]], [2*A[20]+A[ 19]+3*A[18]+2*A[24]+6*A[26]+5*A[23]+6*A[22]+5*A[11]+6*A[10]+5*A[9]+3*A[25]+21*A [8]+7*A[7]+14*A[6]+5*A[21]+35*A[3]+42*A[4]+35*A[5], 2*A[20]+A[19]+3*A[18]+2*A[ 24]+6*A[26]+5*A[23]+6*A[22]+5*A[11]+6*A[10]+3*A[25]+21*A[8]+7*A[7]+14*A[6]+42*A [4]+35*A[5], 3*A[20]+5*A[19]+A[18]+3*A[24]+2*A[26]+4*A[23]+4*A[11]+A[25]+21*A[7 ]+7*A[6]+28*A[5], 4*A[20]+2*A[19]+6*A[18]+4*A[24]+5*A[26]+6*A[25]+35*A[8]+14*A[ 7]+28*A[6], 4*A[20]+2*A[19]+5*A[26]+6*A[25]+35*A[8]+14*A[7], 4*A[20]+5*A[26]+35 *A[8], 14*A[8]], [A[20]+4*A[19]+5*A[18]+A[24]+3*A[26]+6*A[23]+3*A[22]+6*A[11]+3 *A[10]+6*A[9]+5*A[25]+35*A[8]+21*A[7]+21*A[6]+6*A[21]+42*A[3]+21*A[4]+42*A[5], A[20]+4*A[19]+5*A[18]+A[24]+3*A[26]+6*A[23]+3*A[22]+6*A[11]+3*A[10]+5*A[25]+35* A[8]+21*A[7]+21*A[6]+21*A[4]+42*A[5], 5*A[20]+6*A[19]+4*A[18]+5*A[24]+A[26]+2*A 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*A[22]+4*A[10]+A[25]+21*A[7]+7*A[6]+A[21]+7*A[3]+28*A[4]+6*A[30], 5*A[20]+6*A[ 19]+5*A[18]+A[24]+A[26]+4*A[25]+14*A[8]+28*A[7]+21*A[6], A[20]+5*A[19]+3*A[26]+ A[25]+35*A[8]+21*A[7], 5*A[20]+A[26]+35*A[8]], [3*A[9]+4*A[2]+21*A[3]+3*A[30], 6*A[21]+14*A[3]+4*A[30], 2*A[22]+6*A[10]+5*A[21]+7*A[3]+42*A[4]+2*A[30], 5*A[20 ]+6*A[19]+4*A[18]+5*A[24]+A[26]+3*A[23]+4*A[25]+14*A[8]+28*A[7]+21*A[6]+7*A[5], A[20]+4*A[19]+3*A[24]+3*A[26]+5*A[25]+35*A[8]+21*A[7]+7*A[6], 4*A[20]+5*A[26]+3 *A[25]+35*A[8]+7*A[7], 3*A[26]+7*A[8]], [6*A[9]+2*A[2]+42*A[3]+6*A[30], 5*A[9]+ 6*A[21]+42*A[3]+4*A[30], A[22]+3*A[10]+3*A[21]+7*A[3]+21*A[4]+4*A[30], 5*A[23]+ A[22]+A[10]+2*A[21]+7*A[3]+7*A[4]+7*A[5]+5*A[30], A[20]+4*A[19]+3*A[18]+3*A[26] +5*A[25]+35*A[8]+21*A[7]+14*A[6], 4*A[20]+3*A[19]+5*A[26]+35*A[8]+14*A[7], 3*A[ 20]+14*A[8]], [6*A[9]+3*A[2]+42*A[3]+6*A[30], 3*A[9]+2*A[21]+28*A[3]+5*A[30], A [22]+4*A[10]+2*A[21]+7*A[3]+28*A[4]+5*A[30], 2*A[20]+A[19]+3*A[18]+2*A[24]+6*A[ 26]+4*A[23]+2*A[22]+2*A[10]+3*A[25]+21*A[8]+7*A[7]+14*A[6]+4*A[21]+7*A[3]+14*A[ 4]+7*A[5]+3*A[30], 2*A[20]+A[19]+6*A[18]+A[24]+6*A[26]+3*A[25]+21*A[8]+7*A[7]+ 28*A[6], 6*A[20]+6*A[19]+4*A[26]+A[25]+28*A[7], 6*A[20]+A[26]+14*A[8]], [3*A[9] +21*A[3]+3*A[30], 35*A[3]+5*A[9]+5*A[21], A[22]+A[10]+6*A[21]+7*A[3]+7*A[4]+A[ 30], A[22]+A[10]+2*A[21]+7*A[3]+7*A[4]+5*A[30], 4*A[20]+2*A[19]+3*A[18]+2*A[24] +5*A[26]+6*A[25]+35*A[8]+14*A[7]+14*A[6], 3*A[20]+3*A[19]+2*A[26]+2*A[25]+14*A[ 7], 3*A[20]+2*A[26]+21*A[8]], [5*A[9]+A[2]+35*A[3]+5*A[30], 6*A[21]+7*A[3], 3*A [22]+4*A[10]+A[21]+7*A[3]+28*A[4]+6*A[30], 5*A[20]+6*A[19]+4*A[18]+5*A[24]+A[26 ]+6*A[23]+4*A[25]+14*A[8]+28*A[7]+21*A[6]+7*A[5], 4*A[20]+2*A[19]+6*A[24]+5*A[ 26]+6*A[25]+35*A[8]+14*A[7]+7*A[6], 3*A[20]+2*A[26]+6*A[25]+7*A[7], 6*A[26]+14* A[8]], [6*A[9]+4*A[2]+42*A[3]+6*A[30], 3*A[9]+4*A[21]+28*A[3]+2*A[30], 3*A[22]+ A[21]+7*A[3]+7*A[4]+6*A[30], 5*A[20]+6*A[19]+4*A[18]+5*A[24]+A[26]+3*A[23]+2*A[ 22]+2*A[10]+4*A[25]+14*A[8]+28*A[7]+21*A[6]+4*A[21]+7*A[3]+14*A[4]+7*A[5]+3*A[ 30], 6*A[18]+28*A[6], 6*A[20]+6*A[19]+4*A[26]+28*A[7], 6*A[20]+21*A[8]], [3*A[9 ]+6*A[2]+21*A[3]+3*A[30], 6*A[9]+6*A[21]+A[30], 6*A[10]+42*A[4], A[20]+4*A[19]+ 5*A[18]+A[24]+3*A[26]+A[23]+4*A[22]+4*A[10]+5*A[25]+35*A[8]+21*A[7]+21*A[6]+A[ 21]+7*A[3]+28*A[4]+7*A[5]+6*A[30], A[20]+4*A[19]+5*A[18]+2*A[24]+3*A[26]+5*A[25 ]+35*A[8]+21*A[7]+28*A[6], 2*A[20]+5*A[19]+6*A[26]+2*A[25]+21*A[8]+28*A[7], 5*A [20]+2*A[26]+14*A[8]], [6*A[9]+4*A[2]+42*A[3]+6*A[30], 3*A[9]+4*A[21]+28*A[3]+2 *A[30], 3*A[22]+A[21]+7*A[3]+7*A[4]+6*A[30], 5*A[20]+6*A[19]+4*A[18]+5*A[24]+A[ 26]+3*A[23]+2*A[22]+2*A[10]+4*A[25]+14*A[8]+28*A[7]+21*A[6]+4*A[21]+7*A[3]+14*A [4]+7*A[5]+3*A[30], 6*A[18]+28*A[6], 6*A[20]+6*A[19]+4*A[26]+28*A[7], 6*A[20]+ 21*A[8]], [6*A[9]+6*A[2]+42*A[3]+6*A[30], 6*A[9]+3*A[21]+4*A[30], 3*A[22]+2*A[ 10]+6*A[21]+7*A[3]+21*A[4]+A[30], 2*A[20]+A[19]+3*A[18]+2*A[24]+6*A[26]+A[23]+4 *A[22]+4*A[10]+3*A[25]+21*A[8]+7*A[7]+14*A[6]+A[21]+7*A[3]+28*A[4]+7*A[5]+6*A[ 30], 2*A[20]+A[19]+5*A[18]+2*A[24]+6*A[26]+3*A[25]+21*A[8]+7*A[7]+28*A[6], 3*A[ 20]+5*A[19]+2*A[26]+2*A[25]+28*A[7], 5*A[20]+2*A[26]+42*A[8]], [2*A[9]+3*A[2]+ 14*A[3]+2*A[30], 3*A[9]+6*A[21]+28*A[3]+A[30], 4*A[22]+A[21]+7*A[3]+7*A[4]+6*A[ 30], 3*A[20]+5*A[19]+A[18]+3*A[24]+2*A[26]+4*A[23]+2*A[22]+2*A[10]+A[25]+21*A[7 ]+7*A[6]+4*A[21]+7*A[3]+14*A[4]+7*A[5]+3*A[30], 3*A[20]+5*A[19]+6*A[18]+A[24]+2 *A[26]+A[25]+21*A[7]+28*A[6], 6*A[19]+A[25]+28*A[7], 6*A[20]+A[26]+42*A[8]]], [ 1, 1, 3, 13, 14, 27, 17, 22, 31, 41, 42, 48, 38, 43, 2, 39, 0, 11, 23, 31, 11, 29, 28, 43, 25, 6, 14, 7, 14, 39, 8, 28, 15, 46, 41, 30, 18, 0, 32, 44, 17, 24, 34, 42, 6, 45, 22, 11, 0, 32, 9, 10, 28, 15, 11, 34, 11, 13, 38]] Just for fun, , C(100000000000000000000), modulo , 49, is , 0 and the next, 10, terms in the sequence C(n) modulo, 49, are 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ------------------------------------------------ This ends this thrilling article, that took, 3960.749, seconds. to produce .