This way of constructing what is known as the rabbit sequence, I already
gave in
http://www.jsoftware.com/pipermail/programming/2007-December/009177.html,
including repeated squaring:
ts'([:; (;@:{&.><)^:(]`(0 1;~1:)))5'
0.19197681 2.3489024e8
(f9seq 34)-:([:; (;@:{&.><)^:(]`(0 1;~1:)))5
1
ts 'f9seq 34' NB. f9seq from
http://www.jsoftware.com/jwiki/Essays/Fibonacci%20Sequence#RewriteRules
0.7917748 92279552
Before that the elegant solution (,#;._1)^:(]`1:) was given in
http://www.jsoftware.com/pipermail/programming/2007-November/008780.html
ts'|.(,#;._1)^:(]`1:)33'
0.64488748 2.6844032e8
(|.(,#;._1)^:(]`1:)33)-:([:; (;@:{&.><)^:(]`(0 1;~1:)))5
1
R.E. Boss
> -----Oorspronkelijk bericht-----
> Van: [email protected] [mailto:[email protected]] Namens
> Roger Hui
> Verzonden: donderdag 21 mei 2009 1:46
> Aan: Chat forum
> Onderwerp: Re: [Jchat] Silver ratio, geometrically interpreted
>
> Good one.
>
> NB. rule: 1->2, 2->1 2
> Fib =: (#~0&<)@(,/@:({&(>2;1 2)))&:<:^:
> fib1=: -.&0@,@:({&(3 2$0 0 2 0 1 2))^:
> fib2=: ;@({&('';2;1 2))^:
>
> fib1 and fib2 are shorter equivalents to Fib .
>
> NB. rule 0->1 ; 1->0 1
> fib3=: 3 : '+/;@({&(1;0 1))^:y 0' " 0
>
> fib3 i.2 10
> 0 1 1 2 3 5 8 13 21 34
> 55 89 144 233 377 610 987 1597 2584 4181
>
>
>
> ----- Original Message -----
> From: Viktor Cerovski <[email protected]>
> Date: Wednesday, May 20, 2009 16:13
> Subject: Re: [Jchat] Silver ratio, geometrically interpreted
> To: [email protected]
>
> >
> >
> > Tracy Harms-3 wrote:
> > >
> > > A nice bit of illustrated commentary on a special number we
> > don't tend
> > > to hear a lot about:
> > >
> > > http://maxwelldemon.wordpress.com/2009/05/20/unscheduled-post-
> > the-silver-ratio/
> > >
> > Both golden (Fibonacci) and silver (octonacci) numbers can be
> > obtained from
> > substitution rules applied to sequences of a two-letter
> > alphabet. For
> > instance:
> >
> > Fib=:(#~ 0&<)@(,/@:({&(>2;1 2)))&:<:^:
> > NB. rule: 1->2, 2->12
> > 7 Fib 1
> > 2 1 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2
> > +/ 2 = 7 Fib 1
> > 13
> > +/ 2 = 10 Fib 1
> > 55
> > fib =: 3 : '+/2=(y-1)Fib 1'
> >
> > I think that fib is not mentioned in the Roger's essay on
> > generating the Fibonacci's sequence.
> >
> > Octonacci chains can be defined similarly as:
> >
> > Oct=:(#~ 0&<)@(,/@:({&(>2;2 1 2)))&:<:^:
> >
> > 3 Oct 2 1
> > 2 1 2 2 2 1 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2
> >
> > Such chains have many interesting properties...
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