Correcting f(fib(n+1)/fib(n)) and f(fib(n)/fib(n+1)) On Mon, Oct 7, 2013 at 1:31 PM, Bill Gosper <billgosper@gmail.com> wrote:
Some empirical observations: f(n) = 2^n-1 f(1/n) = 2^(n-1) f(1-1/n) = 2^n-2 f(n/(2*n+1)) = 2^(n+2)-4 f(n-1/n) = 1/2 (-2 - 2^n + 2^(2 n)) f(n+1/n) = 1/2 (-2 + 2^(2 n) + 2^(1 + n))
f(fib(n+1)/fib(n)) = 1/12 (-4 + (-2)^n + 9 2^n) f(fib(n)/fib(n+1)) = 1/12 (-8 - (-2)^n + 9 2^n) f(nth convergent(√3)) = 1/56 (-40 + 2^(3 n/2) (13 + 12 Sqrt[2] + (-1)^n (13 - 12 Sqrt[2])))
But for ratios of g[n] = g[n-2]+g[n-3], In[1]:= First /@ NestList[{#[[2]], #[[3]], #[[1]] + #[[2]]} &, {1, 1, 1}, 33]
Out[1]= {1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739}
In[5]:= nam[Numerator[#], Denominator[#]] & /@Ratios[%1]
Out[5]= {1, 1, 3, 1, 5, 9, 17, 25, 49, 9, 9, 497, 9, 8177, 65521, 1048561, 49137, 262129, 327665, 589809, 131057, 1638385, 458737, 3342321, 196593, 7012337, 50528241, 65208305, 1245169, 532873201, 8588034033, 61014001, 2153971697}
(four 9s because of four 4/3s in the ratios) Note that f(fib(n)) = 2^fib(n)-1, a double exponential, but f(fib(n+1)/fib(n)) is not. --rwg
On Sat, Oct 5, 2013 at 9:43 PM, Bill Gosper <billgosper@gmail.com> wrote:
Is anything known about the growth of this function? In[1130]:= NestWhile[{2*Ceiling[1/#[[1]]] - 1 - 1/#[[1]], #[[2]] + 1} &, {3/8, 0}, #[[1]] != 0 &]
Out[1130]= {0, 20}
In[1131]:= NestWhile[{2*Ceiling[1/#[[1]]] - 1 - 1/#[[1]], #[[2]] + 1} &, {22/7, 0}, #[[1]] != 0 &]
Out[1131]= {0, 519}
In[1132]:= NestWhile[{2*Ceiling[1/#[[1]]] - 1 - 1/#[[1]], #[[2]] + 1} &, {355/113, 0}, #[[1]] != 0 &]
Out[1132]= {0, 67107847}
Checks:
In[1133]:= Timing[ NestWhile[{1/(2*Floor[#[[1]]] + 1 - #[[1]]), #[[2]] + 1} &, {0, 0}, #[[1]] != 22/7 &]]
Out[1133]= {0.005219, {22/7, 519}}
In[1134]:= Timing[ NestWhile[{1/(2*Floor[#[[1]]] + 1 - #[[1]]), #[[2]] + 1} &, {0, 0}, #[[1]] != 355/113 &]]
Out[1134]= {700.483896, {355/113, 67107847}} --rwg
On Sat, Oct 5, 2013 at 6:20 PM, Bill Gosper <billgosper@gmail.com> wrote:
In[1106]:= NestList[2*Ceiling[1/#] - 1 - 1/# &, 3/8, 22]
Out[1106]= {3/8, 7/3, 4/7, 5/4, 1/5, 4, 3/4, 5/3, 2/5, 5/2, 3/5, 4/3, 1/4, 3, 2/3, 3/2, 1/3, 2, 1/2, 1, 0, Indeterminate, Indeterminate}
So 3/8 is the "20th rational".
The 68th iterate on √2: In[1127]:= ContinuedFraction[ Nest[Simplify[2*Ceiling[1/#] - 1 - 1/#] &, Sqrt[2], 68]]
Out[1127]= {1, 1, 1, 1, 1, 1, 1, 1, 1, {2}}
I.e., it tries to disguise √2 as the golden ratio by sticking nine 1s on the front. Many other iterates of this process produce CFs with only 1s and 2s. --rwg