> Help18 := proc() print(`IsCond3G(C,G), Cond3G(v,G) , SeqC(N,G), SeqCe(N,G), `); print(`SeqCo(N,G), NuC(v,G) `); end proc;
> with(combinat);
> IsCond3G := proc(C, G) local i, P, T; P := permute(3); for i to nops(C) do T[P[i]] := C[i]; end do; if G[1][2] <= T[[1, 2, 3]] + T[[1, 3, 2]] + T[[3, 1, 2]] - T[[2, 1, 3]] - T[[2, 3, 1]] - T[[3, 2, 1]] and G[2][3] <= T[[1, 2, 3]] + T[[2, 1, 3]] + T[[2, 3, 1]] - T[[1, 3, 2]] - T[[3, 1, 2]] - T[[3, 2, 1]] and -T[[2, 3, 1]] - T[[3, 2, 1]] - T[[3, 1, 2]] + T[[2, 1, 3]] + T[[1, 2, 3]] + T[[1, 3, 2]] < G[1][3] then true; else false; end if; end proc;
> Cond3G := proc(v, G) local A, a, C; A := Comps(v, 6); C := {}; for a in A do if IsCond3G(a, G) then C := C union {a}; end if; end do; C; end proc;
> NuC := proc(v, G) local A, su, a, i; A := Comps(v, 6); su := 0; for a in A do if IsCond3G(a, G) then su := su + add(a)!/mul(a[i]!, i = 1 .. nops(a)); end if; end do; su; end proc;
> SeqCo := proc(N, G) local n; [seq(nops(Cond3G(2*n - 1, G)), n = 1 .. N)]; end proc;
> SeqCe := proc(N, G) local n; [seq(nops(Cond3G(2*n, G)), n = 1 .. N)]; end proc;
> SeqC := proc(N, G) local n; [seq(nops(Cond3G(n, G)), n = 1 .. N)]; end proc;
> Help17 := proc() print(`Comps(n,k), GP1(L,n,d), GP(L,n) , IsCond3(C) `); end proc;
> Comps := proc(n, k) local S, a, S1, s; option remember; if k = 0 then if n = 0 then RETURN({[]}); else RETURN({}); end if; end if; S := {}; for a from 0 to n do S1 := Comps(n - a, k - 1); S := S union {seq([op(s), a], s in S1)}; end do; S; end proc;
> GP1 := proc(L, n, d) local a, i, P, eq, var; if nops(L) < d + 2 then ERROR(`List too short`); end if; P := add(a[i]*n^i, i = 0 .. d); var := {seq(a[i], i = 0 .. d)}; eq := {seq(L[i] = subs(n = i, P), i = 1 .. d + 2)}; var := solve(eq, var); if var = NULL then RETURN(FAIL); end if; P := subs(var, P); if {seq(L[i] - subs(n = i, P), i = d + 3 .. nops(L))} <> {0} then RETURN(FAIL); end if; P; end proc;
> GP := proc(L, n) local d, P; for d from 0 to nops(L) - 2 do P := GP1(L, n, d); if P <> FAIL then RETURN(P); end if; end do; FAIL; end proc;
> IsCond3 := proc(C) local i, v, P, T; P := permute(3); v := add(C); for i to nops(C) do T[P[i]] := C[i]; end do; if 1/2*v < T[[1, 2, 3]] + T[[1, 3, 2]] + T[[3, 1, 2]] and 1/2*v < T[[1, 2, 3]] + T[[2, 1, 3]] + T[[2, 3, 1]] and 1/2*v < T[[2, 3, 1]] + T[[3, 2, 1]] + T[[3, 1, 2]] then true; else false; end if; end proc;
> SeqCoStupid := proc(N) local n; [seq(NuC(2*n - 1, [[0, 1, 1], [0, 0, 1]]), n = 1 .. N)]; end proc;
> L := t -> (1 + t^3*x[1, 2, 3]*x[2, 3, 1]*x[3, 1, 2])*t^3*x[3, 1, 2]*x[1, 2, 3]*x[2, 3, 1]/(1 - t^2*x[1, 2, 3]*x[3, 2, 1])(1 - t^2*x[2, 1, 3]*x[3, 1, 2])(1 - t^2*x[3, 1, 2]*x[1, 2, 3])(1 - t^2*x[1, 3, 2]*x[1, 2, 3])(1 - t^2*x[1, 2, 3]*x[2, 3, 1])(1 - t^2*x[2, 3, 1]*x[3, 1, 2]);
L := proc (t) options operator, arrow; (1+t^3*x[1, 2, 3]*x[2, 



   3, 1]*x[3, 1, 2])*t^3*x[3, 1, 2]*x[1, 2, 3]*x[2, 3, 



   1]/(((((1-t^2*x[1, 2, 3]*x[3, 2, 1])(1-t^2*x[2, 1, 3]*x[3, 



   1, 2]))(1-t^2*x[3, 1, 2]*x[1, 2, 3]))(1-t^2*x[1, 3, 2]*x[1, 



   2, 3]))(1-t^2*x[1, 2, 3]*x[2, 3, 1]))(1-t^2*x[2, 3, 1]*x[3, 



   1, 2]) end proc




;
> taylor(L(t), t = 0, 2);
Error, (in series/function) unable to compute series
;
> NULL;
> Help17 := proc() print(`Comps(n,k), GP1(L,n,d), GP(L,n) , IsCond3(C) `); end proc;
> with(combinat);
> Comps := proc(n, k) local S, a, S1, s; option remember; if k = 0 then if n = 0 then RETURN({[]}); else RETURN({}); end if; end if; S := {}; for a from 0 to n do S1 := Comps(n - a, k - 1); S := S union {seq([op(s), a], s in S1)}; end do; S; end proc;
> GP1 := proc(L, n, d) local a, i, P, eq, var; if nops(L) < d + 2 then ERROR(`List too short`); end if; P := add(a[i]*n^i, i = 0 .. d); var := {seq(a[i], i = 0 .. d)}; eq := {seq(L[i] = subs(n = i, P), i = 1 .. d + 2)}; var := solve(eq, var); if var = NULL then RETURN(FAIL); end if; P := subs(var, P); if {seq(L[i] - subs(n = i, P), i = d + 3 .. nops(L))} <> {0} then RETURN(FAIL); end if; P; end proc;
> GP := proc(L, n) local d, P; for d from 0 to nops(L) - 2 do P := GP1(L, n, d); if P <> FAIL then RETURN(P); end if; end do; FAIL; end proc;
> IsCond3 := proc(C) local i, v, P, T; P := permute(3); v := add(C); for i to nops(C) do T[P[i]] := C[i]; end do; if 1/2*v < T[[1, 2, 3]] + T[[1, 3, 2]] + T[[3, 1, 2]] and 1/2*v < T[[1, 2, 3]] + T[[2, 1, 3]] + T[[2, 3, 1]] and 1/2*v < T[[2, 3, 1]] + T[[3, 2, 1]] + T[[3, 1, 2]] then true; else false; end if; end proc;
> GQP1 := proc(L, p, n) local S, i, j; S := [seq(L[i*p + n], i = 1 .. p)]; GP(S, n); end proc;
> GQP := proc(L, n) local p, S, i; S := []; for i to n do if GQP1(L, i, n) <> FAIL then S := [op(S), GQP1(L, i, n)]; end if; end do; S; end proc;
> GQP([1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14], 3);
                               []


;
> NULL;
> NULL;
> seq(evalf(NuC(2*n - 1, [[0, 1, 1], [0, 0, 1]])/6^(2*n - 1)), n = 1 .. 9);
0., 0.02777777778, 0.03472222222, 0.03750857339, 0.03898807823, 



  0.03990703828, 0.04053420709, 0.04098979008, 0.04133580897




;
> NULL;