• Superseeker is an email server that tries very hard to find an explanation for a number sequence.
• To use it, send an email message to
  superseeker@research.att.com

  containing a line like

  lookup 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
  

• Remarks.
  • Only one request may be submitted at a time.
  • Since this program does some serious computing, only one request per user may be submitted per hour. (There is a list of special users who are exempt from this restriction.)
  • The entries should be separated by spaces rather than commas.
  • The "Subject" line should say "None" or be left blank.
  • Sending an empty message will produce the "Help" file.
  • A web version of the help file is also available, as well as a French version.
• What Superseeker does.
  • Superseeker is an extensive library of programs that tries a great many things to explain the sequence.
  • Some programs try to find a formula or recurrence or rule that directly explains the sequence. These programs were written by Harm Derksen, Olivier Gerard, Christian Kratthentaler, Simon Plouffe, Bruno Salvy, Paul Zimmermann and the author.
  • Other programs apply over 120 different transformations to the given sequence to see if any transformed sequence matches a sequence in the database.
  • See the help file for more details.

The following is an essentially unedited copy of Superseeker's reply to a recent submission from a crystallographer in Switzerland.

The received message said:

lookup 1 10 36 91 190 351

The following is Superseeker's response.

Report on [ 1,10,36,91,190,351]:
Many tests are carried out, but only potentially useful information
(if any) is reported here.


TEST: IS THE NTH TERM A POLYNOMIAL IN N?
        SUCCESS: nth term is nontrivial polynomial in n of degree 4
Polynomial is:
1+15/4*n+31/8*n^2+5/4*n^3+1/8*n^4



Even though there are a large number of sequences in the table, at least
one of yours is not there! Please send it to me using
the submission form on the sequence web page
Submit.html
and I will (probably) add it!  Include a brief description. Thanks!

TRY "RATE", CHRISTIAN KRATTHENTALER'S MATHEMATICA PROGRAM FOR GUESSING 
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)

RATE found the following formula for the nth term:
Warning: as with all these guessing programs, this is only a suggestion!

(n*(3 + n)*(-2 + 3*n + n^2))/8


TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN

        SUCCESS
        (limited to 40 matches):

Transformation T050 gave a match with:
%I A021247
%S A021247 0,0,4,1,1,5,2,2,6,3,3,7,4,4,8,5,5,9,6,7,0,7,8,1,8,9,3,0,0,4,1,1,5,
%T A021247 2,2,6,3,3,7,4,4,8,5,5,9,6,7,0,7,8,1,8,9,3,0,0,4,1,1,5,2,2,6,3,3,7,
%U A021247 4,4,8,5,5,9,6,7,0,7,8,1,8,9,3,0,0,4,1,1,5,2,2,6,3,3,7,4,4,8,5,5,9
%N A021247 Decimal expansion of 1/243.
%K A021247 nonn,cons
%O A021247 0,3
%A A021247 N. J. A. Sloane (njas(AT)research.att.com)

Transformation T019 gave a match with:
%I A009879
%S A009879 1,4,9,17,29,44,62,85,112,139,169,206,247,292,336,380,434,492,548,607,
%T A009879 676,755,832,904,982,1067,1156,1247,1340,1444,1554,1661,1765,1865,
%U A009879 1973,2098,2228,2358,2488,2621,2765,2905,3032,3165,3316,3478,3642,3806
%N A009879 Coordination sequence T5 for Zeolite Code DFO.
%D A009879 R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic 
Description of Coordination Sequences and Exact Topological Densities for Zeolites,
 Acta Cryst., A52 (1996), pp. 879-889.
%D A009879 W.M. Meier, D.H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure 
Types, 4th Ed., Elsevier, 1996
%H A009879 Grosse-Kunstleve et al. paper
%H A009879 Coordination Sequences and Encyclopedia of Integer Sequences
%K A009879 nonn
%O A009879 0,2
%A A009879 rwgk(AT)cci.lbl.gov (R.W. Grosse-Kunstleve)


List of transformations used:
T019  sequence u[j+2]-2*u[j+1]+u[j]
T050  jth coefficient of Sn(z)*(1-z)^j

Abbreviations used in the above 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

Comments

• We can see that the sequence was not in the database, so the sender is asked to submit it to the database for possible inclusion. (This can be done by going to the web page www.research.att.com/~njas/sequences/Submit.html.)
• Two different subroutines were able to find a simple formula for the nth term that matched all the terms that were supplied.
• Two of the transformations succeeded in matching the sequence to sequences already in the database. The first match is obviously irrelevant, but the second is to a certain coordination sequence from zeolite crystallography, and so may have been exactly the sequence that the sender was hoping to find.
• Several of the articles in the list of papers that cite the On-Line Encyclopedia came out of discoveries made with the help of Superseeker.

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