[math-fun] Killing Fibonacci's Rabbits
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence. http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945 Is there a general form for such sequences?
Difficult to know what at level to answer this question ... All these sequences satisfy homogeneous linear recurrences with constant coefficients, about which there is a considerable body of classical theory. Try a web search on "linear recurring sequence" or "linear feedback shift-register sequence"; or see http://en.wikipedia.org/wiki/Recurrence_relation The term-by-term sum and product of such LFSR sequences are also LFSR sequences, but not in general the quotient. BIll Gosper recently asked for an algorithm to detect the quotient of LFSR sequences. I have been waiting for sombody like Neil Sloane or Simon Plouffe to wade in, but in vain ... maybe there's a trade secret to protect? Or maybe they just can't think of anything intelligent to say about it. Like me. WFL On 7/10/13, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote:
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence. http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html
Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945
Is there a general form for such sequences?
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Such algorithms are classical, see for example Massey 1969. Reeds & Sloane generalized the algorithm to handle rings rather than just fields. Charles Greathouse Analyst/Programmer Case Western Reserve University On Wed, Jul 10, 2013 at 5:53 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Difficult to know what at level to answer this question ...
All these sequences satisfy homogeneous linear recurrences with constant coefficients, about which there is a considerable body of classical theory.
Try a web search on "linear recurring sequence" or "linear feedback shift-register sequence"; or see http://en.wikipedia.org/wiki/Recurrence_relation
The term-by-term sum and product of such LFSR sequences are also LFSR sequences, but not in general the quotient. BIll Gosper recently asked for an algorithm to detect the quotient of LFSR sequences. I have been waiting for sombody like Neil Sloane or Simon Plouffe to wade in, but in vain ... maybe there's a trade secret to protect?
Or maybe they just can't think of anything intelligent to say about it. Like me.
WFL
On 7/10/13, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote:
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence.
http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html
Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945
Is there a general form for such sequences?
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Here's a paper by Robert Rumely giving a proof of a result of Alf van der Poorten which characterizes quotients. Victor http://www.math.uga.edu/~rr/HadamardQuotient1.pdf On Wed, Jul 10, 2013 at 5:53 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Difficult to know what at level to answer this question ...
All these sequences satisfy homogeneous linear recurrences with constant coefficients, about which there is a considerable body of classical theory.
Try a web search on "linear recurring sequence" or "linear feedback shift-register sequence"; or see http://en.wikipedia.org/wiki/Recurrence_relation
The term-by-term sum and product of such LFSR sequences are also LFSR sequences, but not in general the quotient. BIll Gosper recently asked for an algorithm to detect the quotient of LFSR sequences. I have been waiting for sombody like Neil Sloane or Simon Plouffe to wade in, but in vain ... maybe there's a trade secret to protect?
Or maybe they just can't think of anything intelligent to say about it. Like me.
WFL
On 7/10/13, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote:
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence.
http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html
Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945
Is there a general form for such sequences?
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"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?) Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder? A search for "dominant root method" yielded numerous hits, including Umberto Zannier (2000) "A proof of Pisot’s d-th root conjecture" Annals of Mathematics, 151 (2000), 375–383 http://arxiv.org/pdf/math/0010024.pdf citing further references. WFL On 7/11/13, Victor Miller <victorsmiller@gmail.com> wrote:
Here's a paper by Robert Rumely giving a proof of a result of Alf van der Poorten which characterizes quotients.
Victor
http://www.math.uga.edu/~rr/HadamardQuotient1.pdf
On Wed, Jul 10, 2013 at 5:53 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Difficult to know what at level to answer this question ...
All these sequences satisfy homogeneous linear recurrences with constant coefficients, about which there is a considerable body of classical theory.
Try a web search on "linear recurring sequence" or "linear feedback shift-register sequence"; or see http://en.wikipedia.org/wiki/Recurrence_relation
The term-by-term sum and product of such LFSR sequences are also LFSR sequences, but not in general the quotient. BIll Gosper recently asked for an algorithm to detect the quotient of LFSR sequences. I have been waiting for sombody like Neil Sloane or Simon Plouffe to wade in, but in vain ... maybe there's a trade secret to protect?
Or maybe they just can't think of anything intelligent to say about it. Like me.
WFL
On 7/10/13, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote:
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence.
http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html
Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945
Is there a general form for such sequences?
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* Fred lunnon <fred.lunnon@gmail.com> [Jul 11. 2013 08:26]:
"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?)
Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder?
http://www.math.uga.edu/~rr/HadamardQuotient2.pdf
[...]
On 11/07/2013 09:05, Joerg Arndt wrote:
* Fred lunnon <fred.lunnon@gmail.com> [Jul 11. 2013 08:26]:
"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?)
Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder?
Did he by any chance also publish an algorithm for reconstructing a mathematical paper given only its odd-numbered pages? -- g
* Gareth McCaughan <gareth.mccaughan@pobox.com> [Jul 11. 2013 12:14]:
On 11/07/2013 09:05, Joerg Arndt wrote:
* Fred lunnon <fred.lunnon@gmail.com> [Jul 11. 2013 08:26]:
"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?)
Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder?
Did he by any chance also publish an algorithm for reconstructing a mathematical paper given only its odd-numbered pages?
Interpolation? If that fails: email Robert Rumely (web search gives nothing, not even the scanned pdf we know about). Best, jj
-- g
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Here's van der Poorten's paper which Rumely just refers to as "to be published": http://maths.mq.edu.au/~alf/www-centre/alfpapers/a077.pdf Victor On Thu, Jul 11, 2013 at 4:38 AM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On 11/07/2013 09:05, Joerg Arndt wrote:
* Fred lunnon <fred.lunnon@gmail.com> [Jul 11. 2013 08:26]:
"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?)
Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder?
Did he by any chance also publish an algorithm for reconstructing a mathematical paper given only its odd-numbered pages?
-- g
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Is there a general form for such sequences?
While contemplating such conundra you might enjoy viewing http://www.supercartoons.net/cartoon/736/whats-opera-doc.html or any of its many internet clones. (It's been deemed "culturally, historically, or aesthetically significant" by the United States Library of Congress, so you must be vewwy vewwy quiet!)
That's one fine cartoon. I'm not an opera, or a Wagner, fan, but I do really like the overture from Tannhäuser (e.g., at < http://www.youtube.com/watch?v=Qx5HL1_h2Fk >) which was used a lot, starting mid-cartoon. I didn't know you could find old cartoons on the Web. I'll be watching a lot of them now -- thanks, Marc! --Dan On 2013-07-11, at 10:12 PM, Marc LeBrun wrote:
Is there a general form for such sequences?
While contemplating such conundra you might enjoy viewing
http://www.supercartoons.net/cartoon/736/whats-opera-doc.html
or any of its many internet clones.
(It's been deemed "culturally, historically, or aesthetically significant" by the United States Library of Congress, so you must be vewwy vewwy quiet!)
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participants (8)
-
Charles Greathouse -
Dan Asimov -
Fred lunnon -
Gareth McCaughan -
Joerg Arndt -
Marc LeBrun -
Stuart Anderson -
Victor Miller