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# Edit detail for SandBox.GuessingSequence revision 1 of 534

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changed:
-
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:

\begin{axiom}
)lib RECOP FAMR2 FFFG FFFGF NEWTON UFPS UFPS1 GOPT GOPT0 GUESS GUESSINT GUESSP GUESSF1 GUESSF
\end{axiom}

See GuessingFormulasForSequences for some explanations.

\begin{axiom}
guess([1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2)
\end{axiom}

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 $q$ being such a rational
function, there is no formula of the form $\prod_{i=0}^nq(i)$ or
$\sum_{i=0}^nq(i)$, nor $\prod_{i_1=0}^n\prod_{i_2=0}^{i_1}q(i_2)$,
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...

\begin{axiom}
guessExpRat [(1+x)^x for x in 0..3]
\end{axiom}

A workaround is necessary, because of bug #128

\begin{axiom}
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)]

guessPRec(q)(l, []).1
\end{axiom}

Here are some that are tried:

\begin{axiom}
listA := [1,1,2,5,14,42,132];
listB := [1,2,6,21,80, 322];
guess(listA, [guessRat], [guessSum, guessProduct])
guess(listB, [guessRat], [guessSum, guessProduct])
\end{axiom}

\begin{axiom}
li :=  [-86, -975, -100, -1728, -31213];
guess(li, [guessRat], [guessSum, guessProduct])
\end{axiom}

\begin{axiom}
guessPRec [1,2,3,4,5,6]
\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:

axiom)lib RECOP FAMR2 FFFG FFFGF NEWTON UFPS UFPS1 GOPT GOPT0 GUESS GUESSINT GUESSP GUESSF1 GUESSF
)library cannot find the file FFFGF.
RecurrenceOperator is now explicitly exposed in frame initial
RecurrenceOperator will be automatically loaded when needed from
/var/zope2/var/LatexWiki/RECOP.NRLIB/code
FiniteAbelianMonoidRingFunctions2 is now explicitly exposed in frame
initial
FiniteAbelianMonoidRingFunctions2 will be automatically loaded when
needed from /var/zope2/var/LatexWiki/FAMR2.NRLIB/code
FractionFreeFastGaussian is now explicitly exposed in frame initial
FractionFreeFastGaussian will be automatically loaded when needed
from /var/zope2/var/LatexWiki/FFFG.NRLIB/code
NewtonInterpolation is now explicitly exposed in frame initial
NewtonInterpolation will be automatically loaded when needed from
/var/zope2/var/LatexWiki/NEWTON.NRLIB/code
UnivariateFormalPowerSeries is now explicitly exposed in frame
initial
UnivariateFormalPowerSeries will be automatically loaded when needed
from /var/zope2/var/LatexWiki/UFPS.NRLIB/code
UnivariateFormalPowerSeriesFunctions is now explicitly exposed in
frame initial
when needed from /var/zope2/var/LatexWiki/UFPS1.NRLIB/code
GuessOption is now explicitly exposed in frame initial
GuessOption will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GOPT.NRLIB/code
GuessOptionFunctions0 is now explicitly exposed in frame initial
GuessOptionFunctions0 will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GOPT0.NRLIB/code
Guess is now explicitly exposed in frame initial
Guess will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GUESS.NRLIB/code
GuessInteger is now explicitly exposed in frame initial
GuessInteger will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GUESSINT.NRLIB/code
GuessPolynomial is now explicitly exposed in frame initial
GuessPolynomial will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GUESSP.NRLIB/code
GuessFiniteFunctions is now explicitly exposed in frame initial
GuessFiniteFunctions will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GUESSF1.NRLIB/code
GuessFinite is now explicitly exposed in frame initial
GuessFinite will be automatically loaded when needed from
/var/zope2/var/LatexWiki/GUESSF.NRLIB/code

See GuessingFormulasForSequences? for some explanations.

axiomguess([1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233], [guessRat], [guessSum, guessProduct], maxLevel==2) (1)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)

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...

axiomguessExpRat [(1+x)^x for x in 0..3] (2)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)

A workaround is necessary, because of bug #128

axioml := [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)] (3)
Type: List Polynomial Integer
axiomguessPRec(q)(l, []).1 (4)
Type: Record(function: Expression Integer,order: NonNegativeInteger?)

Here are some that are tried:

axiomlistA := [1,1,2,5,14,42,132];
Type: List PositiveInteger?
axiomlistB := [1,2,6,21,80, 322];
Type: List PositiveInteger?
axiomguess(listA, [guessRat], [guessSum, guessProduct]) (5)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)
axiomguess(listB, [guessRat], [guessSum, guessProduct]) (6)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)

axiomli :=  [-86, -975, -100, -1728, -31213];
guess(li, [guessRat], [guessSum, guessProduct]) (7)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)

axiomguessPRec [1,2,3,4,5,6] (8)
Type: List Record(function: Expression Integer,order: NonNegativeInteger?)