Alphabet {1,2,3} [1, 2, 3], vs. , [1, 1, 1] using WhoWon [true, .353, .917] Using Findrec and Asy: 2 3 n + 1 2 n N (19 n + 101) N (26 n + 161) N gu := [[- ------------ + ------------ + --------------- - --------------- 6561 (n + 8) 2187 (n + 8) 6561 (n + 8) 2187 (n + 8) 4 5 6 7 (45 n + 191) N (10 n + 57) N 5 (5 n + 24) N (4 n + 37) N + --------------- - -------------- - --------------- - ------------- 729 (n + 8) 243 (n + 8) 81 (n + 8) 27 (n + 8) 8 9 (19 n + 134) N 4 (2 n + 15) N 10 + --------------- - --------------- + N , 9 (n + 8) 3 (n + 8) 25 23 191 529 1469 4085 11381 10589 5 1/2 [1/2, 1/2, --, --, ---, ----, ----, -----, -----, -----], [-------, (1/n) 54 54 486 1458 4374 13122 39366 39366 1/2 8 Pi 1/2 1/2 1/2 365 (1/n) 3324341 (1/n) 951884999 (1/n) + ------------ + ---------------- + ------------------ 64 n 2 3 40960 n 524288 n 1/2 1/2 7393765161867 (1/n) 18101480585293203 (1/n) n + 1 + ---------------------- + --------------------------]], [----------- 4 5 729 (n + 8) 134217728 n 8589934592 n 2 3 4 (n + 3) N (17 n + 61) N (14 n + 47) N (14 n + 75) N + ----------- + -------------- + -------------- + -------------- 243 (n + 8) 729 (n + 8) 243 (n + 8) 81 (n + 8) 5 6 (2 n + 25) N (n + 6) N 7 8 + ------------- - ---------- - N + N , 27 (n + 8) 3 (n + 8) 25 71 203 193 1657 4747 13 1/2 [1/2, 1/2, --, ---, ---, ---, ----, -----], [-------, (1/n) 54 162 486 486 4374 13122 1/2 8 Pi 1/2 1/2 1/2 1577 (1/n) 1976317 (1/n) 2220397115 (1/n) + ------------- + ---------------- + ------------------- 832 n 2 3 106496 n 6815744 n 1/2 1/2 14627870762127 (1/n) 31823790760025943 (1/n) + ----------------------- + --------------------------]]] 4 5 1744830464 n 111669149696 n 2 3 n + 1 2 n N (19 n + 101) N (26 n + 161) N [[- ------------ + ------------ + --------------- - --------------- 6561 (n + 8) 2187 (n + 8) 6561 (n + 8) 2187 (n + 8) 4 5 6 7 (45 n + 191) N (10 n + 57) N 5 (5 n + 24) N (4 n + 37) N + --------------- - -------------- - --------------- - ------------- 729 (n + 8) 243 (n + 8) 81 (n + 8) 27 (n + 8) 8 9 (19 n + 134) N 4 (2 n + 15) N 10 + --------------- - --------------- + N , 9 (n + 8) 3 (n + 8) 25 23 191 529 1469 4085 11381 10589 5 1/2 [1/2, 1/2, --, --, ---, ----, ----, -----, -----, -----], [-------, (1/n) 54 54 486 1458 4374 13122 39366 39366 1/2 8 Pi 1/2 1/2 1/2 365 (1/n) 3324341 (1/n) 951884999 (1/n) + ------------ + ---------------- + ------------------ 64 n 2 3 40960 n 524288 n 1/2 1/2 7393765161867 (1/n) 18101480585293203 (1/n) n + 1 + ---------------------- + --------------------------]], [----------- 4 5 729 (n + 8) 134217728 n 8589934592 n 2 3 4 (n + 3) N (17 n + 61) N (14 n + 47) N (14 n + 75) N + ----------- + -------------- + -------------- + -------------- 243 (n + 8) 729 (n + 8) 243 (n + 8) 81 (n + 8) 5 6 (2 n + 25) N (n + 6) N 7 8 + ------------- - ---------- - N + N , 27 (n + 8) 3 (n + 8) 25 71 203 193 1657 4747 13 1/2 [1/2, 1/2, --, ---, ---, ---, ----, -----], [-------, (1/n) 54 162 486 486 4374 13122 1/2 8 Pi 1/2 1/2 1/2 1577 (1/n) 1976317 (1/n) 2220397115 (1/n) + ------------- + ---------------- + ------------------- 832 n 2 3 106496 n 6815744 n 1/2 1/2 14627870762127 (1/n) 31823790760025943 (1/n) + ----------------------- + --------------------------]]] 4 5 1744830464 n 111669149696 n [[-1/6561*(n+1)/(n+8)+2/2187*n/(n+8)*N+1/6561*(19*n+101)/(n+8)*N^2-1/2187*(26*n +161)/(n+8)*N^3+1/729*(45*n+191)/(n+8)*N^4-1/243*(10*n+57)/(n+8)*N^5-5/81*(5*n+ 24)/(n+8)*N^6-1/27*(4*n+37)/(n+8)*N^7+1/9*(19*n+134)/(n+8)*N^8-4/3*(2*n+15)/(n+ 8)*N^9+N^10, [1/2, 1/2, 25/54, 23/54, 191/486, 529/1458, 1469/4374, 4085/13122, 11381/39366, 10589/39366], [5/8/Pi^(1/2), (1/n)^(1/2)+365/64/n*(1/n)^(1/2)+ 3324341/40960/n^2*(1/n)^(1/2)+951884999/524288/n^3*(1/n)^(1/2)+7393765161867/ 134217728/n^4*(1/n)^(1/2)+18101480585293203/8589934592/n^5*(1/n)^(1/2)]], [1/ 729*(n+1)/(n+8)+1/243*(n+3)/(n+8)*N+1/729*(17*n+61)/(n+8)*N^2+1/243*(14*n+47)/( n+8)*N^3+1/81*(14*n+75)/(n+8)*N^4+1/27*(2*n+25)/(n+8)*N^5-1/3*(n+6)/(n+8)*N^6-N ^7+N^8, [1/2, 1/2, 25/54, 71/162, 203/486, 193/486, 1657/4374, 4747/13122], [13 /8/Pi^(1/2), (1/n)^(1/2)+1577/832/n*(1/n)^(1/2)+1976317/106496/n^2*(1/n)^(1/2)+ 2220397115/6815744/n^3*(1/n)^(1/2)+14627870762127/1744830464/n^4*(1/n)^(1/2)+ 31823790760025943/111669149696/n^5*(1/n)^(1/2)]]] -------------------------------- This took, 31.262, seconds.