Re: [math-fun] World's shortest Fibonacci formula
Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|. In[301]:= Table[Abs@ChebyshevU[n, I/2], {n, -1, 9}] Out[301]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} —rwg On Mon, Jan 7, 2019 at 6:19 AM Bill Gosper <billgosper@gmail.com> wrote:
U_n(i/2)/iⁿ, where U is the special function known as Chebyshev polynomial of the second kind:
In[964]:= Table[ChebyshevU[n, I/2]/I^n, {n, 0, 9}]
Out[964]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55} --Bill U_n(x):= In[173]:= Sin[(n + 1) ArcCos@x]/Sin@ArcCos@x
Out[173]= Sin[(1 + n) ArcCos[x]]/Sqrt[1 - x^2]
In[174]:= FullSimplify@Table[%, {n, 0, 5}]
Out[174]= {1, 2 x, -1 + 4 x^2, -4 x + 8 x^3, 1 - 12 x^2 + 16 x^4, 6 x - 32 x^3 + 32 x^5}
On 20/03/2020 02:06, Bill Gosper wrote:
Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|.
Bah. Even shorter: Fib(n) = F_n. (More explicitly: how can it make sense to call something a "short Fibonacci formula" when it depends on a one-letter notation for something that is both less familiar and requires a longer definition than the Fibonacci numbers themselves?) -- g
My opinion is that Chebyshev polynomials are of more fundamental importance than the Fibonacci sequence, so this definition seems worthwhile. Also, what is the shortest gene sequence that expresses a logarithmic spiral in the morphology of a plant or animal species? I.e., When growing a spiral pattern, is a sunflower somehow more efficient than a pine cone or a nautilus? I don’t know too much about programming DNA, but would be interested to learn more. —Brad
On Mar 19, 2020, at 9:29 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 20/03/2020 02:06, Bill Gosper wrote:
Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|.
Bah. Even shorter: Fib(n) = F_n.
(More explicitly: how can it make sense to call something a "short Fibonacci formula" when it depends on a one-letter notation for something that is both less familiar and requires a longer definition than the Fibonacci numbers themselves?)
-- g
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= Brad Klee <bradklee@gmail.com> wrote:
...Chebyshev polynomials are of more fundamental importance
They are anyhow the "trig parts" of exp n log z which seems pretty core.
...what is the shortest gene sequence that expresses a logarithmic spiral in the morphology
I think the global form is emergent from the basic nature of the local growth processes, rather than being explicitly programmed anywhere. For example the code just says something like "increase tube diameter steadily while keeping in contact with previously grown tube wall". Particularly intriguing is that discontinuities can even arise out of continuity -- Rene Thom's morphogenetic "catastrophes" etc. Perhaps more likely to be explicitly encoded somehow are discrete values -- such as 5-fold symmetry (why is 7-fold so much rarer?)
Proof: In[302]:= Table[ChebyshevU[n, I/2]/I^n, {n, -1, 9}] Out[302]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} In[303]:= Sum[ChebyshevU[n, I/2] t^(n + 1)/I^n, {n, -1, ∞}] Out[303]= (t (-Sqrt[2 (3 + Sqrt[5])] + Sqrt[10 (3 + Sqrt[5])] + 4 t + Sqrt[2 (3 + Sqrt[5])] t^2 - Sqrt[10 (3 + Sqrt[5])] t^2))/(4 (1 - 3 t^2 + t^4)) In[304]:= FullSimplify@% Out[304]= -(t/(-1 + t + t^2)) In[305]:= Series[%,{t,0,9}] Out[305]= t+t^2+2 t^3+3 t^4+5 t^5+8 t^6+13 t^7+21 t^8+34 t^9+O[t]^10 —rwg On Thu, Mar 19, 2020 at 7:06 PM Bill Gosper <billgosper@gmail.com> wrote:
Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|.
In[301]:= Table[Abs@ChebyshevU[n, I/2], {n, -1, 9}]
Out[301]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} —rwg
On Mon, Jan 7, 2019 at 6:19 AM Bill Gosper <billgosper@gmail.com> wrote:
U_n(i/2)/iⁿ, where U is the special function known as Chebyshev polynomial of the second kind:
In[964]:= Table[ChebyshevU[n, I/2]/I^n, {n, 0, 9}]
Out[964]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55} --Bill U_n(x):= In[173]:= Sin[(n + 1) ArcCos@x]/Sin@ArcCos@x
Out[173]= Sin[(1 + n) ArcCos[x]]/Sqrt[1 - x^2]
In[174]:= FullSimplify@Table[%, {n, 0, 5}]
Out[174]= {1, 2 x, -1 + 4 x^2, -4 x + 8 x^3, 1 - 12 x^2 + 16 x^4, 6 x - 32 x^3 + 32 x^5}
participants (4)
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Bill Gosper -
Brad Klee -
Gareth McCaughan -
Marc LeBrun