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THIS IS THE HELP FILE FOR SUPERSEEKER
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Greetings from "superseeker". This program will accept a
sequence of integers and try very hard to find an explanation.
Superseeker makes use of many things, including:
1. The On-Line Encyclopedia of Integer Sequences™ (OEIS™)
- see http://oeis.org/wiki/Welcome
2. The "gfun" Maple package of Bruno Salvy and Paul Zimmermann.
3. A Mathematica program from Olivier Gerard to carry out various
sequence transformations.
4. Harm Derksen's program "guesss", which uses Pade'-Hermite approximations
- see http://www.maplesoft.com/CyberMath/share/guesss.html.
That algorithm is described in:
An algorithm to compute generalized Pade'-Hermite forms,
Report 9403 (1994), Dept. of Math., Catholic University Nijmegen
(available from http://www.math.lsa.umich.edu/~hderksen/preprints/pade.ps).
5. Christian Kratthentaler's Mathematica program Rate which tries to
guess a closed form for a sequence ("rate" is "guess" in German).
For a description of Rate, see
http://radon.mat.univie.ac.at/People/kratt/rate/rate.html.
6. John Linderman's program CheckSeq.pl which checks if
there is a partial overlap between a sequence and any
sequence in the OEIS.
7. Various other programs.
INSTRUCTIONS:
To submit a sequence to superseeker, send mail to
superseeker@oeis.org
containing a single line of the form
lookup 1 2 4 6 10 14 20 26 36 46 60 74 94 114
in the body of the message. In the Subject line, say "none".
The terms must be separated by spaces (not commas).
It is best to give between 10 and 20 terms.
Only one request may be submitted at a time, and (since this
program does some serious computing), only one request per user
per hour. (There is a list of special users who are exempt from
this restriction. If you feel you have a genuine need to be
placed on this list, send mail to president@oeis.org, giving
your reasons!)
[To simply look up sequences in the OEIS, it is
more efficient to use either the email server at
sequences@oeis.org
- send an empty email message to that address to get instructions-
or the Web server at
http://oeis.org/ ]
RECOMMENDATIONS:
. Start from the beginning of the sequence (although of course
keep in mind that different people may define the zero-th
term (say) in different ways
. Minus signs (if any) should be INCLUDED, since most of the
programs will make use of them
. If you receive 30 matches from the OEIS, try again
giving more terms
. If you give too many terms, it will tie up the machine for a long
time, and other users will be very unhappy
. For an array of numbers, try looking up the individual
rows, columns or diagonals, whichever seem appropriate
. The program is (mostly) concerned only with infinite sequences
. The program only handles integers. (For a sequence of
rationals, try the sequences of numerators and
denominators separately.)
. The word "lookup" may only appear once in the message
The "Subject" line is thrown away.
TESTS USED
The program will apply some or all of the following tests.
(The program gives up once it has found a sufficient number
of possible explanations. The more trivial the sequence,
the less time the program will spend studying it.
Also some of these tests are not applicable to all sequences.
Only potentially useful results are reported.)
. Look up in the OEIS both the sequence and the sequence
with the first term omitted.
(The reply will report all matches found, up to a limit of 30.)
. Test if a(n) is a polynomial in n [a(n) denotes the n-th term]
In other words, are the differences of some order constant?
. Test if the differences of some order are periodic.
(Suppose the kth order differences are d(1), ...,d(n).
They are said to be periodic if there is a number p, the period,
with 1 <= p <= n-2, such that d(i) = d(j) whenever i = j (mod p).)
. Test if any row of the difference table of some depth is essentially
constant. This detects such sequences as 4^n - n^4.
(Let the usual difference table be
a(0), a(1), a(2), ...
b(0), b(1), ...
c(0), c(1), ...
....
This is the difference table of depth 1. The table of depth 2
has as top row
a(0), b(0), c(0), ...;
and so on.)
. For a 2-valued sequence, compute the six characteristic sequences
associated with the sequence and look them up in the OEIS.
(Suppose the sequence takes only the values X and Y.
The six characteristic sequences, all equivalent to the original, are:
replace X,Y by 1,2; by 2,1;
the positions of the X's, of the Y's;
the run lengths;
and the derivative, i.e. the positions where the sequence changes.)
. Form the generating functions (g.f.) for the sequence for each of
the following 6 types:
ogf ordinary generating function
egf exponential generating function
revogf reversion of ordinary generating function
revegf reversion of exponential generating function
lgdogf logarithmic derivative of ordinary generating function
lgdegf logarithmic derivative of exponential generating function
and attempt to represent them as rational functions,
hypergeometric series, or the solution to a linear differential
equation with polynomial coefficients.
. Look for a linear recurrence with polynomial coefficients for the
coefficients of the above 6 types of g.f.'s.
. Look for a polynomial equation in y and x for the g.f. y(x) of each
of the above 6 types.
. Apply the transformations listed below to the sequence and look up
the result in the OEIS. Stop when 50 matches have been found.
. Test if the sequence is a Beatty sequence.
(A Beatty sequence is one in which the n-th term is [nz],
where z is irrational. The complementary sequence is [ny],
where 1/x + 1/y = 1. Refs: N. J. A. Sloane,
Handbook of Integer Sequences, 1973, p. 29; R. Honsberger,
Ingenuity in Mathematics, 1970, p. 93. If this is a Beatty
sequence the value given for z will produce the given terms,
but this value of z is very far from being unique.)
LIST OF TRANSFORMATIONS THAT MAY BE APPLIED
Abbreviations used in the following list of transformations:
u[j] = j-th term of the sequence
v[j] = u[j]/(j-1)!
Sn(z) = ordinary generating function
En(z) = exponential generating function
T001 the sequence itself
T002 integers not in the sequence
T003 sequence divided by the gcd of its elements
T004 sequence divided by the gcd of its elements, from the 2nd term
T005 sequence divided by the gcd of its elements, from the 3rd term
T006 elements of odd index in the sequence
T007 elements of even index in the sequence
T008 sequence u[j]/(j-1)!
T009 sequence u[j]*j
T010 sequence u[j]/j!
T011 sequence 2*u[j]
T012 sequence 3*u[j]
T013 coefficients of 1/Sn(z)^2
T014 coefficients of Sn(z)^2
T015 coefficients of 1/Sn(z)
T016 coefficients of Sn(z)*(1+z)/(1-z)
T017 coefficients of Sn(z)*(1-z)/(1+z)
T018 sequence u[j+1]-u[j]
T019 sequence u[j+2]-2*u[j+1]+u[j]
T020 sequence u[j+3]-3*u[j+2]+3*u[j+1]-u[j]
T021 coefficients of Sn(z)/(1-z)
T022 coefficients of Sn(z)/(1-z)^2
T023 coefficients of Sn(z)/(1-z)^3
T024 sequence u[j]+u[j+1]
T025 sequence u[j]+u[j+2]
T026 coefficients of Sn(z)/(1+z)
T027 coefficients of Sn(z)/(1+z^2)
T028 sequence u[j]+u[j+1]+u[j+2]
T029 coefficients of Sn(z)/(1+z+z^2)
T030 sequence u[j+2]-u[j]
T031 coefficients of Sn(z)/(1-z^2)
T032 sequence u[j+2]-u[j+1]-u[j]
T033 coefficients of Sn(z)/(1-z-z^2)
T034 sequence u[j]+j
T035 sequence u[j]+2
T036 sequence u[j]+3
T037 sequence u[j]-j
T038 sequence u[j]-2
T039 sequence u[j]-3
T040 sequence u[j]+1
T041 sequence u[j]-1
T042 coefficients of Sn(z)/(1-z+z^2)
T043 sequence u[j+2]-u[j+1]+u[j]
T044 coefficients of Sn(z)/(1+z-z^2)
T045 sequence u[j]+u[j+1]-u[j+2]
T046 sequence u[j]+2*u[j+1]+u[j+2]
T047 sequence u[j]+3*u[j+1]+3*u[j+2]+u[j+3]
T048 coefficients of Sn(z)/(1+z)^2
T049 coefficients of Sn(z)/(1+z)^3
T050 jth coefficient of Sn(z)*(1-z)^j
T051 jth coefficient of Sn(z)*(1+z)^j
T052 jth coefficient of Sn(z)/(1-z)^j
T053 jth coefficient of Sn(z)/(1+z)^j
T054 coefficients of 1/En(z)^2
T055 coefficients of En(z)^2
T056 coefficients of 1/En(z)
T057 coefficients of En(z)*(1+z)/(1-z)
T058 coefficients of En(z)*(1-z)/(1+z)
T059 sequence v[j+1]-v[j]
T060 sequence v[j+2]-2*v[j+1]+v[j]
T061 sequence v[j+3]-3*v[j+2]-3*v[j+1]+v[j]
T062 coefficients of En(z)/(1-z)
T063 coefficients of En(z)/(1-z)^2
T064 coefficients of En(z)/(1-z)^3
T065 sequence v[j]+v[j+1]
T066 sequence v[j]+v[j+2]
T067 coefficients of En(z)/(1+z)
T068 coefficients of En(z)/(1+z^2)
T069 sequence v[j]+v[j+1]+v[j+2]
T070 coefficients of En(z)/(1+z+z^2)
T071 sequence v[j+2]-v[j]
T072 coefficients of En(z)/(1-z^2)
T073 sequence v[j+2]-v[j+1]-v[j]
T074 coefficients of En(z)/(1-z-z^2)
T075 sequence v[j]+j
T076 sequence v[j]+2
T077 sequence v[j]+3
T078 sequence v[j]-j
T079 sequence v[j]-2
T080 sequence v[j]-3
T081 sequence u[j]+j!
T082 sequence u[j]-j!
T083 coefficients of En(z)/(1-z+z^2)
T084 sequence v[j+2]-v[j+1]+v[j]
T085 coefficients of En(z)/(1+z-z^2)
T086 sequence v[j]+v[j+1]-v[j+2]
T087 sequence v[j]+2*v[j+1]+v[j+2]
T088 sequence v[j]+3*v[j+1]+3*v[j+2]+v[j+3]
T089 coefficients of En(z)/(1+z)^2
T090 coefficients of En(z)/(1+z)^3
T091 jth coefficient of En(z)*(1-z)^j
T092 jth coefficient of En(z)*(1+z)^j
T093 jth coefficient of En(z)/(1-z)^j
T094 jth coefficient of En(z)/(1+z)^j
T095 coefficients of product( 1/(1-z^j)^u[j], j=1..inf )
T096 inverse transform to T095, which was "coefficients of product( 1/(1-z^j)^u[j], j=1..inf )"
T097 coefficients of product( 1/(1-z^j)^v[j], j=1..inf )
T098 inverse transform to T097, which was "coefficients of product( 1/(1-z^j)^v[j], j=1..inf )"
T099 sort terms, remove duplicates
T100 binomial transform: b(n)=SUM C(n,k)a(k), k=0..n
T101 inverse binomial transform: b(n)=SUM (-1)^(n-k)*C(n,k)*a(k), k=0..n
T102 boustrophedon transform (see http://www.research.att.com/~njas/doc/bous.ps)
T103 inverse boustrophedon transform (see http://www.research.att.com/~njas/doc/bous.ps)
T104 Euler transform: define b by 1+SUM b(n)x^n = PRODUCT (1-x^n)^-a(n)
T105 inverse Euler transform: define b by 1+SUM a(n)x^n = PRODUCT (1-x^n)^-b(n)
T106 exponentiate: define b by 1 + EGF_B (x) = exp EGF_A (x)
T107 exponential convolution, expand EGF(x)^2
T108 invert: define b by 1+SUM b(n)x^n = 1/(1 - SUM a(n)x^n)
T109 invert: define b by 1+SUM a(n)x^n = 1/(1 - SUM b(n)x^n)
T110 log: define b by EGF of b = log (EGF of a)
T111 Mobius: define b by b(n)=SUM mu(n/d)*a(d), d divides n
T112 inverse Mobius: define b by b(n)=SUM a(d), d divides n
T113 multiply all except leading terms by 2
T114 Stirling-2 transform: b(n) = SUM S(n,k)a(k), k=0..n
T115 Stirling-1 transform: b(n) = SUM s(n,k)a(k), k=0..n
COMMON ERRORS WHEN USING SUPERSEEKER:
Make sure that the word "lookup" does not appear in the
message in the same line as any non-numeric characters.
Make sure that the lookup line has the form
lookup 1 4 9 16 25 GOOD!
and avoid any lines that say things like:
Subject: lookup please BAD!
Subject: lookup BAD!
To: lookup BAD!
lookup 1,4,9,16,... [NO COMMAS OR DOTS ARE ALLOWED!] BAD!
lookup 1 4 9 16 ? BAD!
In a submission to superseeker the word "lookup" may only
appear once in the entire message.
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THE FORMAT USED IN THE OEIS is described in the web page
http://oeis.org/wiki/Style_Sheet
FOR A SIMPLE AND FAST LOOKUP, try sequences@oeis.org
- send a blank message for instructions
TO PROPOSE A NEW SEQUENCE OR COMMENT use the web page
http://oeis.org/Submit.html
TO LOOKUP SEQUENCES USING THE WEB go to
http://oeis.org/
FOR MORE INFORMATION ABOUT
The On-Line Encyclopedia of Integer Sequences
see the Welcome page
http://oeis.org/wiki/Welcome