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last edited 1 year ago by matteo |
Edit detail for SandBox.GuessingSequence revision 7 of 608
added:
From Thomas Sun Jan 27 03:50:01 -0800 2008
From: Thomas
Date: Sun, 27 Jan 2008 03:50:01 -0800
Subject:
Message-ID: <20080127035001-0800@axiom-wiki.newsynthesis.org>
\begin{axiom}
guessExpRat([1,1,6,54,660,10260,194040,4326840,111177360,3234848400,105135861600]).1.function
\end{axiom}
This page makes test uses of the guessing package by Martin Rubey. Feel free to add new sequences or change the sequences to ones you like to try.
We need to load these packages:
fricas
(1) -> )lib RECOP FAMR2 FFFG FFFGF NEWTON UFPS UFPS1 GOPT GOPT0 GUESS GUESSINT GUESSP GUESSF1 GUESSF
)library cannot find the file RECOP. )library cannot find the file FAMR2. )library cannot find the file FFFG. )library cannot find the file FFFGF. )library cannot find the file NEWTON. )library cannot find the file UFPS1. )library cannot find the file GOPT. )library cannot find the file GOPT0. )library cannot find the file GUESS. )library cannot find the file GUESSINT. )library cannot find the file GUESSP. )library cannot find the file GUESSF1. )library cannot find the file GUESSF.
>> System error: The value 33845 is not of type LIST
See GuessingFormulasForSequences? for some explanations.
fricas
guess([1,4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq1}\left[ \right]](images/5593850768678758914-16.0px.png "\label{eq1}\left[ \right]") | (1) |
Type: List(Expression(Integer))
The answer being an empty list tells us, that there is no
rational function of total degree less than 13, that generates
these numbers. Furthermore, for 
being such a rational
function, there is no formula of the form 
or
, nor 
,
nor replacing the products by sums. In fact, if you look at
Sloane's encyclopedia, you will find a good reason for that: I'd
by very surprised to find such a simple formula for such a family
of objects...
fricas
guessExpRat [(1+x)^x for x in 0..3]
![\label{eq2}\left[{{\left(n + 1 \right)}^{n}}\right]](images/6683871585640527075-16.0px.png "\label{eq2}\left[{{\left(n + 1 \right)}^{n}}\right]") | (2) |
Type: List(Expression(Integer))
A workaround is necessary, because of bug #128
fricas
l := [1,1, 1+q, 1+q+q^2, 1+q+q^2+q^3+q^4, 1+q+q^2+q^3+2*q^4+q^5+q^6, 1+q+q^2+q^3+2*q^4+2*q^5+2*q^6+q^7+q^8+q^9, (1+q^4+q^6)*(1+q+q^2+q^3+q^4+q^5+q^6), (1+q^4)*(1+q+q^2+q^3+q^4+q^5+2*q^6+2*q^7+2*q^8+2*q^9+q^10+q^11+q^12)]
![\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \:{q + 1}, \:{{{q}^{2}}+ q + 1}, \:{{{q}^{4}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{{{q}^{6}}+{{q}^{5}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{{{q}^{9}}+{{q}^{8}}+{{q}^{7}}+{2 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
{{q}^{12}}+{{q}^{11}}+{2 \ {{q}^{10}}}+{2 \ {{q}^{9}}}+{2 \ {{q}^{8}}}+{2 \ {{q}^{7}}}+{3 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+
\
\
\displaystyle
{{q}^{3}}+{{q}^{2}}+ q + 1](images/4636869417498219116-16.0px.png "\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \:{q + 1}, \:{{{q}^{2}}+ q + 1}, \:{{{q}^{4}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{{{q}^{6}}+{{q}^{5}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{{{q}^{9}}+{{q}^{8}}+{{q}^{7}}+{2 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+{{q}^{3}}+{{q}^{2}}+ q + 1}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
{{q}^{12}}+{{q}^{11}}+{2 \ {{q}^{10}}}+{2 \ {{q}^{9}}}+{2 \ {{q}^{8}}}+{2 \ {{q}^{7}}}+{3 \ {{q}^{6}}}+{2 \ {{q}^{5}}}+{2 \ {{q}^{4}}}+
\
\
\displaystyle
{{q}^{3}}+{{q}^{2}}+ q + 1") | (3) |
Type: List(Polynomial(Integer))
fricas
guessPRec(q)(l,[]).1
![\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{{{f \left({n}\right)}\mbox{\rm :}}{{{q \ {f \left({n}\right)}\ {{q}^{n}}}-{f \left({n + 2}\right)}+{f \left({n + 1}\right)}}= 0}}, \:{{f \left({0}\right)}= 1}, \: \right.
\
\
\displaystyle
\left.{{f \left({1}\right)}= 1}\right]](images/1166922280319524024-16.0px.png "\label{eq4}\begin{array}{@{}l}
\displaystyle
\left[{{{f \left({n}\right)}\mbox{\rm :}}{{{q \ {f \left({n}\right)}\ {{q}^{n}}}-{f \left({n + 2}\right)}+{f \left({n + 1}\right)}}= 0}}, \:{{f \left({0}\right)}= 1}, \: \right.
\
\
\displaystyle
\left.{{f \left({1}\right)}= 1}\right]") | (4) |
Type: Expression(Integer)
Here are some that are tried:
fricas
listA := [1,1, 2, 5, 14, 42, 132];
Type: List(PositiveInteger?)
fricas
listB := [1,2, 6, 21, 80, 322];
Type: List(PositiveInteger?)
fricas
guess(listA,[guessRat], [guessSum, guessProduct])
![\label{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{\frac{{4 \ {p_{7}}}+ 2}{{p_{7}}+ 2}}}\right]](images/2251505039735867295-16.0px.png "\label{eq5}\left[{\prod_{
\displaystyle
{{p_{7}}= 0}}^{
\displaystyle
{n - 1}}{\frac{{4 \ {p_{7}}}+ 2}{{p_{7}}+ 2}}}\right]") | (5) |
Type: List(Expression(Integer))
fricas
guess(listB,[guessRat], [guessSum, guessProduct])
![\label{eq6}\left[ \right]](images/3015632402752253779-16.0px.png "\label{eq6}\left[ \right]") | (6) |
Type: List(Expression(Integer))
fricas
li := [-86,-975, -100, -1728, -31213]; guess(li, [guessRat], [guessSum, guessProduct])
![\label{eq7}\left[ \right]](images/8589864719357395348-16.0px.png "\label{eq7}\left[ \right]") | (7) |
Type: List(Expression(Integer))
fricas
guessPRec [1,2, 3, 4, 5, 6]
![\label{eq8}\left[{n + 1}\right]](images/4988308849271365123-16.0px.png "\label{eq8}\left[{n + 1}\right]") | (8) |
Type: List(Expression(Integer))
... --Thomas, Sun, 27 Jan 2008 03:04:15 -0800 reply
fricas
)lib RECOP FAMR2 FFFG FFFGF NEWTON UFPS UFPS1 GOPT GOPT0 GUESS GUESSINT GUESSP GUESSF1 GUESSF
)library cannot find the file RECOP. )library cannot find the file FAMR2. )library cannot find the file FFFG. )library cannot find the file FFFGF. )library cannot find the file NEWTON. )library cannot find the file UFPS1. )library cannot find the file GOPT. )library cannot find the file GOPT0. )library cannot find the file GUESS. )library cannot find the file GUESSINT. )library cannot find the file GUESSP. )library cannot find the file GUESSF1. )library cannot find the file GUESSF.
>> System error: The value 33845 is not of type LIST
fricas
guess([1,1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq9}\left[ \right]](images/8299985760225880582-16.0px.png "\label{eq9}\left[ \right]") | (9) |
Type: List(Expression(Integer))
fricas
guess([1,1, 2, 7, 40, 355, 4720, 91690, 2559980, 101724390], [guessRat], [guessSum, guessProduct], maxLevel==2)
![\label{eq10}\left[ \right]](images/652275695860962513-16.0px.png "\label{eq10}\left[ \right]") | (10) |
Type: List(Expression(Integer))
fricas
guessRec([1,1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600]).1.function
>> Error detected within library code: index out of range
fricas
guessAlg([1,1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600]).1.function
>> Error detected within library code: index out of range
fricas
guessExpRat([1,1, 6, 54, 660, 10260, 194040, 4326840, 111177360, 3234848400, 105135861600]).1.function
>> Error detected within library code: index out of range