Sometime around 18 Nov 2010, Julian derived the general closed form for the continued fraction a,c,d,...,z,a+b,c,d,...,z,a+2b,..., but we (or a disk crash) seem to have misplaced it, despite my nagging false(?) memory that I reported it here. This, derived in a possibly different way, makes a nice toy. fromcf[optional preamble,quasiperiod,variable or {variable, nonzero starting value}] In[481]:= fromcf[{1, 2*k, 1}, k] Out[481]= E In[487]:= fromcf[{2}, {1, 2*k, 1}, {k, 1}] Out[487]= E In[473]:= fromcf[{2*n + 1}, n] Out[473]= Coth[1] In[484]:= fromcf[{4*k, 1}, k] Out[484]= -1 + Cot[1/2] In[486]:= fromcf[{Sqrt[2] - 1, -1/Sqrt[2], Sqrt[2] - 1}, {4*n/3 + 2/3}, n] Out[486]= E^3 Code: In[106]:= cfmat[cf_List] := {Numerator /@ #, Denominator /@ #} &@ Convergents[cf][[{-1, -2}]] In[107]:= cfmat[{a_}] := {{a, 1}, {1, 0}} In[108]:= cfmat[{}] = IdentityMatrix[2] Out[108]= {{1, 0}, {0, 1}} Clear[fromcf]; fromcf[pre_List: {}, {a___, (c1_: 1)*v_ + c0_: 0}, {v_Symbol, lo_: 0}] := fromcf[pre, {a, c1*v + c1*lo + c0}, v] /; FreeQ[{a}, v] In[103]:= fromcf[pre_List: {}, {a___, (c1_: 1)*v_ + c0_: 0, b__}, {v_Symbol, lo_: 0}] := fromcf[Join[pre, {a, c1*lo + c0}], {b, a, c1*v + c0 + c1}, {v, lo}] /; FreeQ[{a, b}, v] In[104]:= fromcf[pre_List: {}, {a___, (c1_: 1)*v_ + c0_: 0, b__}, v_Symbol] := fromcf[Join[pre, {a, c0}], {b, a, c1*v + c0 + c1}, v] /; FreeQ[{a, b}, v] fromcf[pre_List, {a___, (c1_: 1)*v_ + c0_: 0}, v_] := Block[{M = cfmat[{a}], A, B, C, D, det}, det = Det[{{A, B}, {C, D}} = M]; FromContinuedFraction[ Append[pre, A/(C + (det Hypergeometric0F1[1 + (B + C + A c0)/(A c1), det/( A^2 c1^2)])/((B + C + A c0) Hypergeometric0F1[( B + C + A c0)/(A c1), det/(A^2 c1^2)]))]]] /; FreeQ[{a}, v] --rwg