Shanks's first order (single transient snuffing) sequence extrapolator: shanks[L_List] := (#1*#3 - #2^2)/(#1 - 2*#2 + #3) &[Drop[L, -2], L[[2 ;; -2]], Drop[L, 2]] (Cf (= californium) http://en.wikipedia.org/wiki/Shanks_transformation .) I.e., In[214]:= shanks[{a, b, c}] Out[214]= {(-b^2 + a c)/(a - 2 b + c)} In[215]:= % == shanks[{c, b, a}] Out[215]= True (Doesn't know frontwards from backwards.) E.g. In[216]:= 69 - 2^Range[9] Out[216]= {67, 65, 61, 53, 37, 5, -59, -187, -443} In[217]:= shanks[%] Out[217]= {69, 69, 69, 69, 69, 69, 69} But Fibonaccis have two transients: In[220]:= Fibonacci[Range[17]] Out[220]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597} In[221]:= shanks[%] Out[221]= {1, ComplexInfinity, 1, -1, 1/2, -(1/3), 1/5, -(1/ 8), 1/13, -(1/21), 1/34, -(1/55), 1/89, -(1/144), 1/233} In[222]:= shanks[%] During evaluation of In[222]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. >> During evaluation of In[222]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered. >> Out[222]= {Indeterminate, Indeterminate, -(1/7), -(1/28), -(1/123), -( 1/515), -(1/2192), -(1/9269), -(1/39291), -(1/166396), -(1/ 704935), -(1/2986039), -(1/12649248)} In[223]:= -1/Drop[%, 2] Out[223]= {7, 28, 123, 515, 2192, 9269, 39291, 166396, 704935, \ 2986039, 12649248} This is A220361 Fibonacci(n)^3 + (-1)^n*Fibonacci(n-2). 1, 7, 28, 123, 515, 2192, 9269, 39291, 166396, 704935, 2986039, 12649248, 53582777, 226980767, 961505180, 4073002563, 17253513691, 73087060144, 309601749709, 1311494066355, 5555578003196, 23533806098447, 99690802365743, 422297015611968, 1788878864731825 (list; graph; refs; listen; history; text; internal format) COMMENTS An integral pentagon is a pentagon with integer sides and diagonals. There are two types of such pentagons. Type A have sides A066259(n+1), A220360(n+1), A066259(n+1), A220360(n+1), A066259(n+1), and opposite diagonals A056570(n+2), A056570(n+2), A220361(n+2), A056570(n+2), A056570(n+2), for n=1,2,... REFERENCES R. K. (Some) Guy, Unsolved Problems in Number Theory, D20. --rwg Using NeilB's nifty general Shanksmaker, In[225]:= Shanks[lis_] := Block[{delta = (Rest[#] - Drop[#, -1]) &[lis], len = Length[lis], k = (Length[lis] - 1)/2, mat}, (mat = Table[delta[[i ;; i + k]], {i, 1, k}]; Det[Prepend[mat, lis[[1 ;; k + 1]]]]/ Det[Prepend[mat, ConstantArray[1, k + 1]]])] construct the two-transient snuffer In[235]:= FullSimplify[Shanks[Slot /@ Range[5]]] Out[235]= (#3^3 + #1 #4^2 + #2^2 #5 - #3 (2 #2 #4 + #1 #5))/(#2^2 + 3 #3^2 + #4^2 - 2 #2 (#3 + #4 - #5) - #1 (#3 - 2 #4 + #5) - #3 (2 #4 + #5)) Shamelessly copying and pasting, In[264]:= shanks2[L_] := (#3^3 + #1 #4^2 + #2^2 #5 - #3 (2 #2 #4 + #1 \ #5))/(#2^2 + 3 #3^2 + #4^2 - 2 #2 (#3 + #4 - #5) - #1 (#3 - 2 #4 + #5) - #3 (2 #4 + #5)) &[ L[[1 ;; -5]], L[[2 ;; -4]], L[[3 ;; -3]], L[[4 ;; -2]], L[[5 ;; -1]]] In[265]:= shanks2[%220] Out[265]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} Annihilates the Fibonaccis. (I thought I knew how to SetDelayed@@{...} to use a computed definition. Does another speaker of this grotesque language wish to ruin the puzzle for me?)