Re: [math-fun] Generalized Fibonacci sequences and arithmetic progressions
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n". —Dan ----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
Sorry; I meant any two-sided sequence (indexed by Z) in which every term is the sum of the two preceding terms. Jim On Tuesday, August 8, 2017, Dan Asimov <dasimov@earthlink.net> wrote:
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n".
—Dan
----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
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I assume it means f(n) + f(n + 1) = f(n + 2) for all n in Z. Andy On Aug 9, 2017 06:02, "Dan Asimov" <dasimov@earthlink.net> wrote:
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n".
—Dan
----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
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Yes, exactly (for instance the Lucas sequence ...,1,3,4,7,11,18,...). Jim On Wednesday, August 9, 2017, Andy Latto <andy.latto@gmail.com> wrote:
I assume it means f(n) + f(n + 1) = f(n + 2) for all n in Z.
Andy
On Aug 9, 2017 06:02, "Dan Asimov" <dasimov@earthlink.net <javascript:;>> wrote:
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n".
—Dan
----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
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Since the classic Fibonacci sequence avoids 4 and 6 mod 8, it follows that any generalized Fibonacci will also miss some values mod 8. In the original sequence mod 8 we have 11235055271011. Now consider a generalized Fibonacci starting with a,b mod 8. If this sequence ever hits 0 mod 8, we have four consecutive terms that look like c,0,c,c. But this means that we have (a shift of) the original sequence multiplied by c, since the original sequence contains 1,0,1,1. But this gives only 6 distinct values mod 8. It feels like this must be a special case of something much more general, but I do not see it at the moment ... On Wed, Aug 9, 2017 at 1:07 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, exactly (for instance the Lucas sequence ...,1,3,4,7,11,18,...).
Jim
On Wednesday, August 9, 2017, Andy Latto <andy.latto@gmail.com> wrote:
I assume it means f(n) + f(n + 1) = f(n + 2) for all n in Z.
Andy
On Aug 9, 2017 06:02, "Dan Asimov" <dasimov@earthlink.net <javascript:;>> wrote:
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n".
—Dan
----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
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Nice! Thanks. Jim On Thursday, August 10, 2017, Michael Collins <mjcollins10@gmail.com> wrote:
Since the classic Fibonacci sequence avoids 4 and 6 mod 8, it follows that any generalized Fibonacci will also miss some values mod 8. In the original sequence mod 8 we have 11235055271011. Now consider a generalized Fibonacci starting with a,b mod 8. If this sequence ever hits 0 mod 8, we have four consecutive terms that look like c,0,c,c. But this means that we have (a shift of) the original sequence multiplied by c, since the original sequence contains 1,0,1,1. But this gives only 6 distinct values mod 8.
It feels like this must be a special case of something much more general, but I do not see it at the moment ...
On Wed, Aug 9, 2017 at 1:07 AM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Yes, exactly (for instance the Lucas sequence ...,1,3,4,7,11,18,...).
Jim
On Wednesday, August 9, 2017, Andy Latto <andy.latto@gmail.com <javascript:;>> wrote:
I assume it means f(n) + f(n + 1) = f(n + 2) for all n in Z.
Andy
On Aug 9, 2017 06:02, "Dan Asimov" <dasimov@earthlink.net <javascript:;> <javascript:;>> wrote:
Would you be so kind as to define what a "two-sided generalized Fibonacci sequence means"? Sorry, I'm too dumb to be able to figure out what the general term is from "n-m,m,n,m+n".
—Dan
----- Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? -----
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participants (4)
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Andy Latto -
Dan Asimov -
James Propp -
Michael Collins