I mentioned f(nth convergent(√3)) = 1/56 (-40 + 2^(3 n/2) (13 + 12 Sqrt[2] + (-1)^n (13 - 12 Sqrt[2]))) where f is Newman's rational counter. Why didn't I say f(1 + (-2 (1 - Sqrt[3])^(-1 + n) + 2 (1 + Sqrt[3])^(-1 + n))/(-(1 - Sqrt[3])^n + (1 + Sqrt[3])^n)) ? Because instead of this formula, I got the puzzling expression In[83]:= FullSimplify[ FindSequenceFunction[Convergents[Sqrt[3], 22]]@n, n \[Element] Integers] Out[83]= (2 Sqrt[3] (-2 - Sqrt[3])^n QPochhammer[2 - Sqrt[3], -2 + Sqrt[3], n])/ ((-1 + (-2 - Sqrt[3])^n) QPochhammer[-1, -2 + Sqrt[3], n]) (Strangely, the supposedly equivalent FullSimplify[FindSequenceFunction[Convergents[Sqrt[3], 22], n], n \[Element] Integers] simply gives up.) OK, let's cut it a break and do the numerators and denoms separately: In[84]:= FullSimplify[ FindSequenceFunction[Numerator /@ #, n]/ FindSequenceFunction[Denominator /@ #, n] &@ Convergents[Sqrt[3], 22], n \[Element] Integers] Out[84]= (3 (2 + Sqrt[ 3]) ((2 - Sqrt[3])^( n/2) (724 Sqrt[2 + Sqrt[3]] + 418 Sqrt[ 3 (2 + Sqrt[3])] + (989 + 571 Sqrt[3]) (-2 + (-1)^n (2 + Sqrt[2]))) + (2 + Sqrt[3])^( n/2) (1978 + 1142 Sqrt[3] + 724 Sqrt[2 + Sqrt[3]] + 418 Sqrt[3 (2 + Sqrt[3])] - (-1)^ n Root[16 + 31648 #1 + 7824944 #1^2 - 7912 #1^3 + #1^4 &, 3])))/((2 + Sqrt[3])^( n/2) ((2 + Sqrt[2]) (6393 + 3691 Sqrt[3]) + (-1)^ n Root[144 + 613728 #1 + 326963664 #1^2 + 51144 #1^3 + #1^4 &, 2]) - 4 I^n (2 - Sqrt[3])^( n/2) (Cos[(n \[Pi])/2] Root[9 - 81740892 #1^2 + #1^4 &, 4] + I (6393 + 3691 Sqrt[3]) Sin[(n \[Pi])/2])) Ingratitude. BtW, In[87]:= Convergents[1/(Sqrt[3] - 1), 9] Out[87]= {1, 3/2, 4/3, 11/8, 15/11, 41/30, 56/41, 153/112, 209/153} I didn't know that In[88]:= NestList[1 + 1/2/# &, 1, 8] Out[88]= {1, 3/2, 4/3, 11/8, 15/11, 41/30, 56/41, 153/112, 209/153} I'm a little surprised that neither this formula nor the first one above requires any odd-even machinery, since the continued fraction of √3 has period 2: 1 1 2 1 2 1 2 ... --rwg