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## [Axiom-developer] [Guessing formulas for sequences]

 From: kratt6 Subject: [Axiom-developer] [Guessing formulas for sequences] Date: Sun, 20 Mar 2005 09:52:49 -0600

```Changes
http://page.axiom-developer.org/zope/mathaction/GuessingFormulasForSequences/diff
--

??changed:
-\Rate, Doron Zeilberger's program 'GuessRat' and the relevant parts of Bruno
'Rate', Doron Zeilberger's program 'GuessRat' and the relevant parts of Bruno

--removed:
-

??changed:
-  \$f(n)-f(n-1)\$, and
\$f(n)-f(n-1)\$, and

??changed:
-    Guessing formulas for sequences of rational numbers
-
-      For example, if we suspect that a sequence of integers or rationals like
Guessing formulas for sequences of rational numbers

For example, if we suspect that a sequence of integers or rationals like

??changed:
-  guessPade(n, [1, 1, 2, 3, 5], n+->n)\\$GuessInteger
guessPade(n, [1, 1, 2, 3, 5], n+->n)\\$GuessInteger

??changed:
-    Guessing formulas for sequences of rational functions
-
-      Most of the previous section still applies, the only change being that
we now
Guessing formulas for sequences of rational functions

Most of the previous section still applies, the only change being that we
now

??changed:
-  guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat],
-        [guessSum, guessProduct])\\$GuessPolynomial
guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat],
[guessSum, guessProduct])\\$GuessPolynomial

??changed:
-Sequence (4)::
Sequence (4) ::

??changed:
-    Some remarks
-
-      - All of the presented guessing algorithms are insensitive to the shift
-        operator. I.e., if a formula for the sequence \$(s_1,s_2,\dots,s_m)\$
can be
-        guessed, then the corresponding formula for \$(s_2,s_3,\dots,s_{m+1})\$
will
-        be found, too.
-
-      - Except of the class of functions covered by 'guessExpRat', all are
-        closed under addition. I.e., if formulas for \$(s_1,s_2,\dots,s_m)\$ and
-        \$(t_1,t_2,\dots,t_m)\$ can be guessed, then also for
-        \$(s_1+t_1,s_2+t_2,\dots,s_m+t_m)\$. However, the class of functions of
-        type (5) is not even closed under addition of a constant! On the other
-        hand, all classes are closed under multiplication.
-
-      - Note that the class of functions covered by 'guessPRec', i.e., the
-        class of \$D\$-finite functions, is closed under the operator \$\Delta_n\$.
-        Thus, it does not make to try to guess a function for some sequence \$s\$
-        with::
-
-[9 more lines...]
Some remarks

- All of the presented guessing algorithms are insensitive to the shift
operator. I.e., if a formula for the sequence \$(s_1,s_2,\dots,s_m)\$ can be
guessed, then the corresponding formula for \$(s_2,s_3,\dots,s_{m+1})\$ will
be found, too.

- Except of the class of functions covered by 'guessExpRat', all are
closed under addition. I.e., if formulas for \$(s_1,s_2,\dots,s_m)\$ and
\$(t_1,t_2,\dots,t_m)\$ can be guessed, then also for
\$(s_1+t_1,s_2+t_2,\dots,s_m+t_m)\$. However, the class of functions of
type (5) is not even closed under addition of a constant! On the other
hand, all classes are closed under multiplication.

- Note that the class of functions covered by 'guessPRec', i.e., the
class of \$D\$-finite functions, is closed under the operator \$\Delta_n\$.
Thus, it does not make sense to try to guess a function for some
sequence \$s\$ with::

guess(n, s, n+->n, [guessPRec], [guessSum]).

- The situation is very different, if the operator 'guessProduct' is
specified, too. The
class of functions covered by::

guess(n, s, n+->n, [guessPRec], [guessSum, guessProduct])

- is bigger than the class of functions covered by::

guess(n, s, n+->n, [guessPRec], [guessProduct])!

--